Hp-BEM for Contact Problems and Extended Ms-FEM in Linear Elasticity

hp-BEM for Contact Problems and Extended Ms-FEM in Linear Elasticity

Von der Fakult¨at fur¨ Mathematik und Physik der Gottfried Wilhelm Leibniz Universit¨at Hannover zur Erlangung des Grades Doktor der Naturwissenschaften Dr. rer. nat.

genehmigte Dissertation

von

Dipl.- Math. Abderrahman Issaoui

geboren am 03.02.1977 in Metlaoui (Tunesien)

2014 Referent: Prof. Dr. Ernst P. Stephan, Leibniz Universit¨at Hannover Korreferent: Prof. Dr. Joachim Gwinner, Universit¨at der Bundeswehr, Munchen¨ Tag der Promotion: 05.06.2014 Abstract

We consider a contact problem between an elastic body and a rigid foundation with Tresca’s friction law, present a hp-discretization technique of a mixed formulation based on biorthogonal basis functions and solve the corresponding discrete system with the semi-smooth Newton method. A-posteriori error estimates using an error indicator in an hp-adaptive refinement algorithm are derived. We show convergence of the hp- version of BEM and present some numerical experiments. Furthermore we consider a mixed boundary element formulation, which is stabilized following ideas of P.Hild, Y. Renard and V.Lleras for the FEM. A mesh-dependent stabilization term is added to the discrete mixed formulation, in order to avoid the discrete inf-sup condition. Exis- tence and uniqueness of the solution of the discrete problem are shown together with a priori error and a posteriori error estimates. Numerical experiments are listed for the stabilized and the non-stabilized cases. A stochastic contact problem with a boundary integral formulation is analyzed. We show that the stochastic mixed formulation is well-posed, study the deformation of an elastic homogeneous material in which Young’s modulus (parameter that characterizes the material properties) is a random variable. An extended multiscale finite element method EMsFEM is derived for the analysis of linear elastic heterogeneous materials. The main idea is to construct numerically finite element basis functions that captures the small-scale information (the fine mesh) within each coarse element. The construction of the basis functions is done separately for each coarse element with piecewise linear functions. The boundary conditions for the construction of the multiscale basis functions have a big influence on capturing the small-scale information. We analyse a corresponding FEM/BEM coupling and derive an a priori error and a-posteriori error estimate. Next we present finite element implementations for nonperiodic case.

Keywords. Tresca frictional contact problem, biorthogonal basis functions, stabilized hp-BEM, semi-smooth Newton, Stochastic BEM, Multiscale-FEM.

I Zusammenfassung

Wir betrachten ein Kontaktproblem mit Trescareibung zwischen einem elastischen Körper und einem starren Untergrund. Fur¨ die entsprechende gemischte Formulierung präsentieren wir eine hp-Diskretisierungstechnik basierend auf biorthogonalen Basisfunktionen und lösen das entsprechende diskrete System mit einem halbglatten Newton-Verfahren. A posteriori Fehlerabschätzungen erlauben die Definition eines Fehlerindikators und eine hp-adaptive Verfeinerungsstrategie. Wir zeigen die Konvergenz einer hp-Version der BEM und präsentieren numerische Experimente. Daruber¨ hinaus betrachten wir eine gemischte Randelementformulierung, die entsprechend den Ideen von P. Hild, V. Ller- as and Y. Renard fur¨ FEM stabilisiert wird. Ein gitter-abhängiger Stabilisierungsterm wird zur diskreten gemischten Formulierung addiert, um die diskrete inf-sup-Bedingung zu vermeiden. Existenz und Eindeutigkeit der Lösung des diskreten Problems werden zusammen mit a priori und a posteriori Fehlerabschätzungen gezeigt. Numerische Ex- perimente fur¨ den stabilisierten und den nicht stabilisierten Fall bestätigen die theore- tischen Ergebnisse. Zudem wird ein stochastisches Kontaktproblem in einer Randinte- gralformulierung analysiert. Wir zeigen, dass die stochastische gemischte Formulierung wohlgestellt ist, und studieren die Verformung eines homogenen elastischen Materials, in dem das Young-Modul (das die Materialeigenschaften charakterisiert) eine Zufallsva- riable ist. Eine erweiterte Multiskalen-Finite-Element-Methode EMsFEM wird fur¨ die Analyse von heterogenen linearen Materialien hergeleitet. Die Hauptidee besteht dar- in, numerisch Finite-Element-Basisfunktionen zu konstruieren, die die Mikrostruktur in jedem Grobelement erfassen. Die Konstruktion der Basisfunktionen wird fur¨ jedes Grobelement mit stckweise linearen funktionen durchgefuhrt.¨ Die Randbedingungen fur¨ die Konstruktion der Mehrskalen-Basisfunktionen haben nämlich einen großen Einfluss auf die Erfassung der Mikrostruktur. Wir analysieren eine entsprechende FEM/BEM- Kopplung sowie a priori und a posteriori Fehlerabschtzungen. Weiter präsentieren wir Finite-Element-Implementierungen fur¨ den nicht periodischen Fall.

Schlagworte: Tresca Reibungskontakt, biorthogonale Basisfunktionen, stabilisiertes hp-BEM, halbglattes Newton-Verfahren, stochastische BEM, Mehrskalen-FEM

II Acknowledgments

First of all, I would like to express my sincere gratitude to my advisor

Prof. Dr. Ernst P. Stephan for the continuous support of my Ph.D study and research, for his patience, motivation, enthusiasm, and immense knowledge.

I have furthermore to thank my co-referee Prof. Dr. Joachim Gwinner for his readiness to examine my thesis.

My special thanks go to Dr. Lothar Banz, Dr. Heiko Gimperlein, and my friend Zouhair Nezhi for their support and help in all stages of this thesis.

I also thank all of the members of the Graduiertenkolleg for the various support.

I wish to extend my thanks to the whole staﬀ at the Institute for Applied Mathematics. Particularly to Mrs. Carmen Gatzen and Mrs. Ulla Fleischhauer for their kindness and their support on technical issues.

I am also very grateful to my brothers Chokri, Imed, Taouﬁk, and to my sisters Zohra and Hanen, and my most heartfelt thanks to my brother Adel Aissaoui, my belle-soeur noussa. Most especially, I would like to thank my wife Imen for her support and patience during my work in this thesis.

I thank my brother Adel for having been the heart of the family and taking such good care of everybody.

Last but not least, I would like to thank my parents for their spiritual care and protection, and for their amazing love and support.

This thesis was supported by IRTG 1627 ( International Research and Training Group, granted by DFG).

Hannover, den 09/04/2014 Abderrahman Issaoui

III

Contents

1 Introduction 1

2 hp-BEM For Frictional Contact Problem in Linear Elasticity 3 2.1 Mixed Formulation for Frictional Contact Problem ...... 3 2.2 Existence and uniqueness of the solution ...... 7 2.3 Discretization of the Lam´eproblem ...... 8 2.4 Semi-smooth Newton method and algebraic representation ...... 12 2.5 A priori error estimate ...... 18 2.6 A posteriori error estimates for contact with friction ...... 29 2.7 Numerical Experiments ...... 33

3 Stabilized mixed hp-BEM in Linear Elasticity 41 3.1 The mixed formulation ...... 41 3.2 The stabilized mixed hp-BEM formulation ...... 43 3.2.1 Existence and uniqueness of the solution ...... 44 3.3 A priori error analysis for frictional contact problem ...... 47 3.4 Reliable and eﬃcient a posteriori error estimates for stabilized hp-BEM for frictional contact problems ...... 64 3.4.1 Reliability of the BEM a posteriori error estimate ...... 64 3.4.2 Eﬃciency of the BEM a posteriori error estimate ...... 71 3.5 Numerical Experiments ...... 79

4 hp-BEM for Stochastic Contact Problems in Linear Elasticity 83 4.1 Mixed Formulation for Stochastic Contact Problem ...... 83 4.2 Discretization for Stochastic Contact Problem ...... 87 4.3 A posteriori error estimates for stochastic contact with friction . . . . . 95 4.4 Numerical Experiments ...... 99

5 Extended Multiscale Finite Element Method in Linear Elasticity 101 5.1 The equation of linear elasticity ...... 101 5.2 The finite element discretization ...... 102 5.3 Extended multiscale finite element method for the analysis of linear elastic heterogeneous materials ...... 105 5.3.1 The construction of the basis functions ...... 106 5.3.2 Stiffness matrix on the coarse mesh ...... 108 5.4 Convergence of the multiscale finite element method ...... 109

V Contents

5.5 A posteriori error for multiscale ﬁnite element method ...... 113 5.6 Coupling FEM-BEM ...... 119 5.6.1 The weak formulation of the Model Problem ...... 120 5.6.2 The discrete problem for Ms-FEM-BEM ...... 121 5.6.3 A priori error estimate ...... 121 5.6.4 A posteriori error ...... 122 5.7 Numerical Experiments ...... 130

VI List of Figures

2.1 Deformed geometry ...... 36 2.2 Estimated errors ...... 36 2.3 Visualization of Lagrange multiplier ...... 37 2.4 Adaptivety generated meshes for Lam´e-BEM(bubble indicator) . . . . . 38 2.5 Adaptivety generated meshes for Lam´e-BEM(residual indicator) . . . . 39

3.1 Initial(left) and Deformed (right) geometry ...... 81 3.2 Convergence for stabilized and non-stabilized problems ...... 81

4.1 Deformed geometry with the mean value of Young modulus E ...... 100 4.2 Convergence of Stochastic Contact problem ...... 100

5.1 [54]The construction of the numerical basis functions ...... 107 x x 5.2 Displacement ﬁeld of T for the basis function Φp1x (left) and Φp1y (right) 107 5.3 [54] Schematic description of the EMsFEM ...... 108 5.4 The distributions of the Mises stress obtained by the EMsFEM with

Mc × Nc = 12 × 2, Mf × Nf = 16 × 16, H = 0.5, h = 0.031250 . . . . . 132 5.5 The distributions of the Mises stress obtained by the EMsFEM with

Mc × Nc = 24 × 4, Mf × Nf = 8 × 8, H = 0.25, h = 0.031250 ...... 132 5.6 The distributions of the Mises stress obtained by the EMsFEM with

Mc × Nc = 48 × 8, Mf × Nf = 4 × 4, H = 0.125, h = 0.031250 . . . . . 132 5.7 The distributions of the Mises stress obtained by the EMsFEM with

Mc × Nc = 96 × 16, Mf × Nf = 2 × 2, H = 0.062500, h = 0.031250 . . 132 5.8 The distributions of the Mises stress obtained by FEM with h = 0.031250 132 5.9 The Young’s modulus and the Poisson’s ratio for the heterogeneous model 133 5.10 Deformed geometry ...... 133 5.11 The distributions of the Mises stress obtained by the EMsFEM with

Mc × Nc = 96 × 16, H = 0.062500, Lh = h = 0.031250 ...... 133 5.12 The distributions of the Mises stress obtained by FEM with Lh = 0.031250 and h = 0.0039 (Lh = 8h) ...... 133

VII

1 Introduction

Frictional contact problems in linear elasticity play an important role in engineering and structural mechanics. In most cases, contact problems are reformulated in terms of variational inequality problems, see Glowinski et al.[28], Kikuchi and Oden [43]. The approximation of contact problems can be realized by the mixed finite element method in which the non-penetration condition and the friction law are only weakly enforced by a variational inequality. The theory of the mixed finite element method is proposed by Haslinger et al.[33],[34] and developed by Babuˇska [6], [7] and Brezzi [14]. The key of this approach is the inf-sup condition. The finite element spaces for the primal variables and the multipliers have to fullfil inf-sup condition, which is needed in the convergence analysis. However, a discrete inf-sup condition is difficult to obtain for the hp-BEM. Solving frictional contact problems with the mixed hp-BEM boundary element method is a challenging task in mechanics.

To circuvent this difficulty a stabilization technique is used, by adding supplementary terms in the weak formulation, this method has been introduced and analyzed by Bar- bosa and Hughes in [11] and [12]. This approach was taken in [36, 37, 45] in the context of h-FEM and is extended to hp-BEM in the present work. The great advantage of this approach is that the stabilized scheme is stable for arbitrary finite element discretizations, since for the Lagrange multiplier space any discretization can be chosen. The stabilized method for low-order finite element discretizations is based on linear H1- conform functions for the displacement and piecewise linear (or constant) functions for the Lagrange multipliers, see [36, 37, 45]. The use of higher-order polynomials leads to a certain non-conformity in the discretization which requires attention in the convergence analysis.

Many multiscale ﬁnite element methods MsFEM have been developed and studied for the analysis of linear elastic heterogeneous materials. The MsFEM has been originally indroduced by Babuˇska et al. [4, 5] and was developed further by Hou et al. [38, 39] for solving second order elliptic boundary value problems.

In this thesis, an extended multiscale ﬁnite element method EMsFEM is derived for the analysis of linear elastic heterogeneous materials. We analyse a FEM/BEM coupling for EMsFEM and derive a priori and a posteriori error estimates .

The thesis is organized as follows. In Chapter 1, frictional contact problems in linear elasticity with Tresca friction in 2D are presented and analyzed. A higher-order hp-BEM discretization thechnique of a mixed formulation based an biorthogonal basis functions

1 1 Introduction is introduced. The use of the biorthogonality allows a componentwise decoupling of the inequality constraints. On the other hand the decoupled contact conditions can be represented by the problem of ﬁnding the root of a non-linear complementarity function, and therefore the so-called semi-smooth Newton methods can be applied. An a priori error analysis is carried out in the case, where we assume the discrete inf-sup condition.

A posteriori error estimates using an error indicator in the hp-adaptive reﬁnement algorithm are derived. We show convergence of the hp-version of BEM and present numerical experiments that illustrate and conﬁrm our theoretical results.

In Chapter 2, we consider a hp-BEM stabilized mixed boundary element formulation for frictional contact problems in linear elasticity in 2D. Here a mesh dependent stabilization term is added to the discrete mixed formulation, in order to avoid the inf-sup condition. The contact constraints are imposed on the discrete global set of aﬃnely transformed Gauss-Lobatto points on the elements. Furthermore, a priori error estimates are presented and a posteriori estimates are derived for higher-order boundary element methods. Numerical experiments are listed for the stabilized and the non- stabilized cases.

In Chapter 3, a stochastic contact problem with boundary integral formulation is analyzed. A stochastic mixed formulation is shown to be well-posed. This problem is transformed into an equivalent deterministic one by using a Karhunen-Lo`eve spectral decomposition. The biorthogonality is adapted in the context of a stochastic hp-BEM . The componentwise decoupling of the weak constraints allows to use an Uzawa algorithm to solve the discrete problem. Here the Uzawa algorithm uses a pointwise projection of the Lagrange multiplier. The deterministic formulation allows us to obtain a residual a posteriori error estimator.

Finally in the last chapter, an extended multiscale finite element method EMsFEM is presented for the analysis of linear elastic heterogeneous materials. The idea of the method is to construct numerically the multiscale basis functions to capture the fine scale features of the coarse elements in the multiscale finite element analysis. The construction is done separately for each coarse element by solving a subgrid problem together with suitable boundary conditions.

We introduce a FEM/BEM coupling for the EMsFEM with standard BEM and derive an a posteriori error estimate. Finally, we give for the EMsFEM some numerical experiments.

2 2 hp-BEM For Frictional Contact Problem in Linear Elasticity

In this chapter, we consider a contact problem between an elastic body and a rigid foundation with Tresca’s friction law and present a hp-discretization technique of a mixed formulation based on biorthogonal basis functions. For standard h-version BEM and Signorini conditions see [32]. We solve the corresponding discrete system with the semi-smooth Newton method, we derive a posteriori error estimates and use the error indicator in an hp-adaptive reﬁnement algorithm. We show convergence for hp-version of BEM and present some numerical experiments. Related a posteriori error estimates for friction problems have been considered, for example, in [27, 30, 36, 42, 46, 50].

2.1 Mixed Formulation for Frictional Contact Problem

d Let Ω ⊂ R , d = 2, 3, be a bounded Lipschitz domain with the boundary Γ := ∂Ω = ΓN ∪ ΓD ∪ ΓC be decomposed into non-intersecting Neumann ΓN , Dirichlet ΓD and contact boundaries ΓC , where ΓC can come in contact with the rigid foundation. The problem then consists in ﬁnding the displacement ﬁelds u :Ω −→ R such that

− div σ(u) = 0 in Ω (2.1a) σ(u) = C : (u) in Ω (2.1b)

u = 0 on ΓD (2.1c)

σ(u) · n = t on ΓN (2.1d)

σn ≤ 0, un ≤ g, σn(un − g) = 0 on ΓC (2.1e)

|σt| ≤ F, σtut + F|ut| = 0 on ΓC (2.1f)

Let G(x,y) be the fundamental solution of the Lam´eequation

 λ+3µ 1 λ+µ (x−y)⊗(x−y)  {log I + 2 }, for d = 2,  4πµ(λ+2µ) |x−y| λ+3µ |x−y| G(x, y) :=   λ+3µ 1 λ+µ (x−y)⊗(x−y) 8πµ(λ+2µ) { |x−y| I + λ+3µ |x−y|3 }, for d = 3,

3 2 hp-BEM For Frictional Contact Problem in Linear Elasticity

We consider a homogenous, isotropic, linear Saint Venant-Kirchhoﬀ material , where the stress tensor is given in term of Hooke’s tensor by

σ(u) = C : ε(u) := λtrε(u)I + 2µε(u)

1 ε(u) = (∇u + ∇uT ) 2 where λ,µ are the Lam´econstants, moreover, tr denotes the matrix trace operator and d I the identity in R , d = 2, 3 The scalar normal and tangential boundary stresses

σn := n · σ(u) · n and σt := t · σ(u) · t are well deﬁned, where n denotes the unit normal exterior to the contact boundary ΓC .

Let us denote by F > 0 the given friction threshold on ΓC , we assume that F is a constant for the sake of simplicity.

The static Tresca friction condition reads as follows

 |σt| ≤ F, a.e on ΓC  if |σt| < F then ut = 0  if |σt| = F then there exists ν > 0 such that ut = −νσt

We introduice the single layer potential V, the double layer potential K, the adjoint double layer potential K 0 and the hypersingular integral operator W for x ∈ Γ by Z V φ(x) := G(x, y)φ(y) dsy, Γ Z T Ku(x) := (Tny G(x, y) )u(y) dsy, Γ 0 Z K φ(x) := Tnx G(x, y)φ(y) dsy, Γ Z T W u(x) := −Tnx (Tny G(x, y) )u(y) dsy Γ where Tn is the boundary traction operator given by

Tn(u) := σ(u)|Γ· n (2.2)

Lemma 2.1. [22, 23] Let Γ := ∂Ω be the boundary of a Lipschitz domain Ω.Then the integral operators

− 1 +s 1 +s 1 +s 1 +s V : H 2 (Γ) → H 2 (Γ),K : H 2 (Γ) → H 2 (Γ) 0 − 1 +s − 1 +s 1 +s − 1 +s K : H 2 (Γ) → H 2 (Γ),W : H 2 (Γ) → H 2 (Γ).

4 2.1 Mixed Formulation for Frictional Contact Problem

1 1 are bounded for all s ∈ [− 2 , 2 ],i.e. there exists constants CV ,CK ,CK0 ,CW > 0 such that

0 kV φk 1 ≤ CV kφk 1 , kK φk 1 ≤ C 0 kφk 1 , H 2 +s(Γ) H− 2 +s(Γ) H− 2 +s(Γ) K H− 2 +s(Γ)

kKuk 1 ≤ CK kuk 1 , kW uk 1 ≤ CW kuk 1 . H 2 +s(Γ) H 2 +s(Γ) H− 2 +s(Γ) H 2 +s(Γ)

d Lemma 2.2. [51, 19] Let Γ := ∂Ω ⊂ R be boundary of a Lipschitz domain Ω.Let − 1 cap(Ω) < 1 in case d = 2. Then the single layer potential V is H 2 (Γ)-elliptic,i.e. there exists a constant cV > 0, such that

− 1 hV φ, φiΓ ≥ cV kφk 1 , ∀φ ∈ H 2 (Γ) (2.3) H− 2 (Γ)

−1 − 1 1 Moreover its inverse operator V : H 2 (Γ) → H 2 (Γ) exists and

−1 −1 − 1 kV uk 1 ≤ c kuk 1 , ∀u ∈ H 2 (Γ), (2.4) H− 2 (Γ) V H 2 (Γ) where cV is the ellipticity constant of V.

The Steklov-Poincar´eoperator S is deﬁned by 1 1 S := W + (K0 + )V −1(K + ). 2 2

1 Lemma 2.3. [19] Let Γ0 ⊂ Γ. Then the Steklov-Poincar´eoperator S : H 2 (Γ) → 1 1 − 2 H 2 (Γ) is continuous and H˜ (Γ0)-elliptic, i.e. there exists cS, CS > 0 such that

1 kSuk 1 ≤ CSkuk 1 ∀u ∈ H 2 (Γ), (2.5) H− 2 (Γ) H 2 (Γ) 1 2 ˜ 2 hSu, uiΓ ≥ cSkuk 1 ∀u ∈ H (Γ0) (2.6) 2 H˜ (Γ0)

For any displacement u and for any boundary traction σ(u)n deﬁned on ∂Ω the following notations are frequently used in this chapter.

u = unn + utt and σ(u)n = σn(u)n + σt(u)t (2.7)

We need the following function spaces

1 1 d 1 V := [H˜ 2 (Σ)] = H˜ 2 (Σ) := {u ∈ H 2 (Γ); supp(u) ⊂ Σ}

1 2 V := {u ∈ H˜ (Σ) : u = 0 on ΓD} 1 d 1 W := [H 2 (ΓC )] = H 2 (ΓC ) − 1 d − 1 M := [H 2 (ΓC )] = H 2 (ΓC ) where

Σ := ΓC ∪ ΓN , ΓC ∩ ΓN = ∅

5 2 hp-BEM For Frictional Contact Problem in Linear Elasticity

Let K be a closed convex and nonempty subset of V, to model the nonpenetration condition on the contact boundary ΓC , where g ≥ 0 is a given gap function.

1 2 K := {u ∈ H˜ (Σ) : un ≤ g on ΓC }

2 − 1 1 We assume that F ∈ L (ΓC ), t ∈ H 2 (ΓN ) and g ∈ H 2 (ΓC ).

We deﬁne the DtN (Dirichlet-to-Neumann) mapping uΓ → σ(u)· n, Since S is the DtN mapping, there holds

σn ≡ Su · n|ΓC , σt ≡ Su · t|ΓC (2.8) where u, σn, σt solve (2.1) in the distributional sense.

We deﬁne the space of Lagrange multipliers by

M = Mn × Mt (2.9) where

1 1 ˜ − 2 2 Mn := {µn ∈ H (ΓC ): hµn, vniΓC ≤ 0, ∀v ∈ H (ΓC ) with vn ≤ 0 a.e on ΓC } and

Mt(F) := {µt ∈ L2(ΓC ): |µt|≤ F a.e on ΓC } are the sets of normal and tangential Lagrange multipliers.

The classical formulation (2.1) can be rewritten in a weak sence as a saddle point problem as follows:

Find (u, λ) ∈ V × M such that

hSu, viΣ + b(λ, v) = ht, viΓN ∀v ∈ V (2.10a) b(µ − λ, u) hg, µ − λ i ∀ µ ∈ M. (2.10b) 6 n n ΓC Here

b(µ, v) := hµ , v i + hµ , v i (2.11) n n ΓC t t ΓC

1 1 − 2 ˜ 2 where the notation h·, ·iΓC represents the duality pairing between H (ΓC ) and H (ΓC ).

Note λ = −σ(u) · n|ΓC ≥ 0 since σ(u) · n ≤ 0 on ΓC in (2.1). Another classical weak formulation of the problem (2.1) is the primal variational inequality:

Find u ∈ K such that

hSu, v − uiΣ + j(v) − j(u) ≥ ht, v − uiΓN ∀v ∈ K (2.12)

6 2.2 Existence and uniqueness of the solution where Z j(u) = F|ut| ds (2.13) ΓC is the friction functional.

As a minimization problem, it reads

J(u) ≤ J(v) ∀v ∈ K (2.14) where 1 J(v) = hSv, vi + j(v) − ht, vi (2.15) 2 Σ ΓN

2.2 Existence and uniqueness of the solution

In this section, we study the existence and uniqueness of the solution of the mixed formulation

Theorem 2.1. There exists a unique solution of problem (2.10).

Proof. We know from [19] that the minimization problem (2.14) and the variational problem (2.12) are equivalent. According to [24] , K is a closed, convex set, and J : K −→ R is continuous, convex and coercive, i.e.

J(v) −→ ∞, for v ∈ K and kvk 1 −→ ∞. H˜ 2 (Σ)

This implies the existence of a minimizer u of (2.14). Furthermore from [24] J is strictly convex. Therefore the minimizer u is unique.

Note that Z

j(v) = F|vt| ds = sup hµt, vtiΓC . (2.16) ΓC µt∈Mt(F)

We deﬁne the Lagrange functional

1 L(v, µ , µ ) := hSv, vi − L(v) + hµ , v − gi + hµ , v i (2.17) n t 2 n n ΓC t t ΓC Note that by (2.16) we get

J(u) = inf sup L(v, µn, µt) = L(v, λn, λt). (2.18) v∈V (µn,µt)∈Mn×Mt(F)

Thus, for any saddle point (u, λn, λt) ∈ V × Mn × Mt(F) of L, u is a minimizer of J.

7 2 hp-BEM For Frictional Contact Problem in Linear Elasticity

Since Mt(F) is bounded, the existence of a unique saddle point is guaranteed, if there exists a constant β > 0 such that

c βkµnk − 1 ≤ sup hµn, γn(v)i (2.19) H 2 (ΓC ) v∈V,kvk 1 =1 H˜ 2 (Σ) where c γn(v) = vn|ΓC The above condition (inf-sup condition ) follows from the closed range theorem and the c surjectivity of γn, see [50] .

We conclude that there exists a unique saddle point (u, λn, λt) ∈ V × Mn × Mt(F) of L, which is equivalently characterized by the mixed variational formulation (2.10).

2.3 Discretization of the Lam´eproblem

Let Thp denote a partition of Γ¯C ∪ Γ¯N into disjoint straight line segments I, such that all corners of Γ¯C ∪ Γ¯N and all end points Γ¯C ∩ Γ¯N , Γ¯D ∩ Γ¯N are nodes of Thp. For simplicity we assume meas(ΓC ) > 0 and Γ¯C ∩ Γ¯D = ∅. Furthermore we deﬁne the set of Gauss-Lobatto points GI,hp on each element I ∈ Thp of corresponding polynomial degree pI and set Ghp := ∪I∈Thp GI,hp, where p = (pI )I∈Thp associates to each element of Thp a polynomial degree pI ≥ 1.

We introduce the space of continuous piecewise polynomials for the discretization of u:

1 1 hp ˜ 2 hp 2 hp 2 Vhp := {u ∈ H (Σ) : ∀I ∈ Thp, u |I ∈ [PpI (I)] , u = 0 on ΓD} ⊂ H (Γ) and the space of piecewise polynomial discrete boundary tractions

1 1 − 2 2 − 2 Whp := {φ ∈ H (Σ) : ∀I ∈ Th, φ|I ∈ [PpI −1(I)] } ⊂ H (Γ)

The space Vhp is spanned by the 2-d nodal basis {φiek, i = 1, ..., NV, k = 1, 2}, where ek denotes the k-th unit vector, φi the scalar Gauss-Lobatto Lagrange basis function associated with the node i and NV the total number of the nodes. We denote by NC the set of all nodes on the contact boundary ΓC . Furthermore we deﬁne the space of discrete vectorial Lagrange multipliers by Mhp, which is spanned by {ψiek, i = 1, ..., NC , k = 1, 2}.

The dual or biorthogonal basis functions ψi satisfy the orthogonality relations Z Z ψiφjds = δij φjds. (2.20) ΓC ΓC In order to obtain a well-deﬁned normal we introduce the averaged normal at the node in ΓC ∩ Ghp as follows.

8 2.3 Discretization of the Lam´eproblem

Let us deﬁne by Ei for i ∈ NC all surface elements of the mesh containing node i .

Ei = {ei ∈ Thp : ei ∈ ΓC , i ∈ NC } (2.21)

Then the normal ni can be deﬁned via 1 X ni := P nei , i ∈ NC (2.22) e ∈E nei i i ei∈Ei

where nei is the unit normal vector of the surface edge ei.

hp We can then express the discrete function u ∈ Vhp for 1 ≤ i ≤ dim Vhp by hp X u := (uinni + uit)φi. (2.23) i The normal and tangential part of the discrete function uhp are given by

hp X hp X un := uinφi, ut = uitφi, (2.24) i i where uin and uit are the normal and tangential component of the nodal value ui, given by

uin := uini, uit := ui − uinni. (2.25)

We deﬁne in the same way the discrete Lagrange multiplier for 1 ≤ i ≤ dim Mhp(F) as hp X λ := (λinni + λit)ψi (2.26) i

For the frictionless contact problem the tangential component λit equals 0.

We introduce the subset

− hp hp Vhp := {v ∈ Vhp : vn ≤ 0}. (2.27) As in [41] we deﬁne the space for the discrete Lagrange multiplier by

hp NC hp hp hp hp − Mhp(F) := {µ ∈ span{ψi}i=1 : hµ , v i ≤ hF, |vt |hi, v ∈ Vhp}, (2.28) hp where the absolute value of the function vt is given for 1 ≤ i ≤ dim Mhp(F) and x ∈ ΓC by (see.[41]) hp X |vt |h := |vit|φi(x). (2.29) i

We deﬁne the weighted gap vector for 1 ≤ i ≤ dim Mhp(F) by 1 Z gi := gψi(x) ds, (2.30) Di ΓC where Z Di = φi(x) ds. ΓC

− The spaces Vhp and Mhp(F) can be rewritten as follows; cf. [[41], Lemma 2.3] for the h-version, i.e. p = 1.

9 2 hp-BEM For Frictional Contact Problem in Linear Elasticity

− Lemma 2.4 ([8], Lemma 6.1). The space Vhp in (2.27) can be rewritten as

− hp X Vhp := {v = viφi ∈ Vhp : vin ≤ 0, i ∈ NC } (2.31) i∈NC and the space in (2.28) is equivalent to

hp X Mhp(F) := {µ = µiψi : µin ≥ 0, |µit| ≤ F, i ∈ NC } (2.32) i∈NC

Since an explicit representation of V −1 is not known, we need to approximate the Steklov-Poincar´eoperator.

− 1 1 Let ihp : Whp ,→ H 2 (Γ) and jhp : Vhp ,→ H 2 (Γ) denote the canonical imbedding the ∗ ∗ dual ihp and jhp

The approximation Shp of the Poincar´e-Steklov operator is given by 1 1 S := W + (K0 + )i (i∗ V i )−1i∗ (K + ) (2.33) hp 2 hp hp hp hp 2

We can rewrite the space Mhp(F) as Mhp(F) := Mn,hp × Mt,hp(F), where Mn,hp and Mt,hp are the sets of normal and tangential discrete Lagrange multiplier, respectively.

The discretized version of (2.10) reads as:

hp hp hp hp Find u ∈ Vhp and λ = (λn , λt ) ∈ Mhp(F) := Mn,hp × Mt,hp(F)

hp hp hp hp hp hp hShpu , v iΣ + b(λ , v ) = L(v ) ∀v ∈ Vhp (2.34a)

hp hp hp D hp hpE hp b(µ − λ , u ) 6 g, µn − λn ∀ µ ∈ Mhp(F). (2.34b) ΓC

d Lemma 2.5. [19] Let Γ := ∂Ω ⊂ R be the boundary of a Lipschitz domain Ω and arbi- 1 trary Γ0 ⊂⊂ Γ.Then the approximation of the Steklov-Poincar´eoperator Shp : H 2 (Γ) → 1 1 − 2 ˜ 2 H (Γ) is continuous and H (Γ0)-elliptic, i.e. there exists cShp , CShp > 0 such that

1 kShpuk 1 ≤ CS kuk 1 ∀u ∈ H 2 (Γ), (2.35) H− 2 (Γ) hp H 2 (Γ) 1 2 ˜ 2 hShpu, uiΓ ≥ cShp kuk 1 ∀u ∈ H (Γ0). (2.36) 2 H˜ (Γ0)

We deﬁne the operator Ehp = S − Shp which represents the error in the approximation of the Steklov-Poincar´eoperator.

Lemma 2.6. [19] The operator Ehp is bounded, i.e. there exists CEhp > 0 such that

1 kEhpuk 1 ≤ CE kuk 1 ∀u ∈ H 2 (Γ), (2.37) H− 2 (Γ) hp H 2 (Γ)

10 2.3 Discretization of the Lam´eproblem

and there exists a constant C0 > 0, such that

− 1 1 kEhpuk 1 ≤ C0 inf kV (K + )u − Φk 1 ∀u ∈ H 2 (Γ). (2.38) − 2 − 2 H (Γ) Φ∈Whp 2 H (Γ)

Extending the approach in [8] for the scalar case to the vector case we obtain the following result. Theorem 2.2. There exists exactly one solution to the discrete mixed formulation (2.34).

hp hp hp hp Proof. Uniqueness: Let (u1 , λ1 ) and (u2 , λ2 ) two solutions of the discrete mixed formulation (2.34). Then we have

hp hp hp hp hp hp hp hp hShp(u1 − u2 ), u1 − u2 iΣ + b(λ1 − λ2 , u1 − u2 ) = 0 (2.39)

hp hp hp hp Choosing µ1 = λ2 and µ2 = λ1 in (2.34b) we get hp hp hp hp b(λ1 − λ2 , u1 − u2 ) ≥ 0 (2.40) Using Lemma 2.5 we obtain

hp hp 2 hp hp hp hp hp hp hp hp cShp ku1 − u2 k 1 ≤ hShp(u1 − u2 ), u1 − u2 iΣ + b(λ1 − λ2 , u1 − u2 ) = 0. H˜ 2 (Σ) (2.41)

Consequently the ﬁrst argument uhp is unique.

hp hp Since u is unique we have for all v ∈ Vhp

hp hp hp 0 = b(λ1 − λ2 , v ). (2.42)

hp hp hp Using the linear combinations of λ1 ,λ2 and v and the biorthogonality, we obtain the relation X 0 = [(λ1,pn − λ2,pn)vpn + (λ1,pt − λ2,pt)vpt]Dp. (2.43)

p∈ΓC ∩Ghp where Z Dp = φp(x) ds. ΓC

Due to the arbitrary choice of vpn and vpt, choosing vpt = 0 in (2.43) we get

(λ1,pn − λ2,pn)vpn = 0 and λ1,pn = λ2,pn

hp We choose now vpn = 0, we obtain λ1,pt = λ2,pt, consequently the second argument λ is unique .

Existence: The inequality (2.34b) is obviously equivalent to the following conditions:

hp hp hp hp hp λn ∈ Mn,hp, hµn − λn , un − gi ≤ 0, ∀µn ∈ Mn,hp, (2.44) hp hp hp hp hp λt ∈ Mt,hp, hµt − λt , ut i ≤ 0, ∀µt ∈ Mt,hp. (2.45)

11 2 hp-BEM For Frictional Contact Problem in Linear Elasticity

The inequalities can be replaced by projections PMn,hp and PMt,hp respectively, where

hp hp hp hp hp hp λn = PMn,hp (λn + r(un − g)), λt = PMt,hp (λt + rut ) (2.46)

Here the maps PMn,hp and PMt,hp stand for the L2 projections onto Mn,hp and Mt,hp, respectively, and r > 0 is an arbitrary parameter. The discrete mixed formulation leads to a ﬁxed points formulation

T : Mn,hp × Mt,hp(F) −→ Mn,hp × Mt,hp(F) hp hp hp hp hp hp (λn , λt ) 7−→ PMn,hp (λn + r(un − g)),PMt,hp (λt + rut ) (2.47) where T is the fixed point operator ,which defines the fixed point iteration

hp k+1 hp k+1 hp k hp k (λn ) , (λt ) = T (λn ) , (λt ) . (2.48)

From Lemma 2.5 we have

hp hp 2 hp hp 2 −1 hp hp hp hp −ku1 − u2 kL (Γ )& −ku1 − u2 k 1 ≥ −cS hShp(u1 − u2 ), u1 − u2 iΣ 2 C H˜ 2 (Σ) hp (2.49)

hp hp hp hp hp hp Using the notation δu = u1 − u2 and δλ = λ1 − λ2 kT (λhp) − T (λhp)k2 ≤ kδλhp + rδuhp| k2 +kδλhp + rδu |hp k2 1 2 L2(ΓC ) n n ΓC L2(ΓC ) t t ΓC L2(ΓC )

≤ kδλhpk2 +2rb(δλhp, δuhp) + r2kδuhpk2 L2(ΓC ) L2(ΓC )

≤ kδλhpk2 −2rhS δuhp, δuhpi + r2kδuhpk2 L2(ΓC ) hp Σ L2(ΓC )

≤ kδλhpk2 −2rc kδuhpk2 +r2kδuhpk2 L2(ΓC ) Shp L2(ΓC ) L2(ΓC )

≤ kδλhpk2 (1 − 2rc β2 + r2β2) L2(ΓC ) Shp

kδλhpk2 where β = L2(ΓC ) and T is strict contraction for 0 < r < 2c . By the Banach kδuhpk2 Shp L2(ΓC ) hp hp hp hp fixed point theorem exists a λ = (λn , λt ) which satisfies (2.47). For any given λ , problem (2.34a) reduces to a linear, finite dimentional problem. Hence, the uniqueness result of uhp implies the existence of a uhp(λhp).

2.4 Semi-smooth Newton method and algebraic representation

For the solution of the discrete mixed formulation (2.34) we describe a semi-smooth Newton approach which is equivalent to an active set strategy.

12 2.4 Semi-smooth Newton method and algebraic representation

Lemma 2.7. The variational inequality constraint (2.34b) is equivalent to the decoupled pointwise non-penetration conditions

uin ≤ gi, λin ≥ 0, λin(uin − gi) = 0 (2.50) and the friction conditions

|λit| ≤ Fi

|λit| < Fi ⇒ uit = 0 (2.51)

|λit| = Fi ⇒ ∃α ∈ R : λit = αuit for all 1 ≤ i ≤ NC where Fi is the friction at the node deﬁned by Z Fi := Fφi ds (2.52) ΓC

Proof. see [41] and [8]

The penalized Fischer-Burmeister non linear complementarity function presented in [18] is deﬁned by q hp hp 2 2 ϕµ(u |ΓC , λ ) = µ λin + (gi − uin) − λin + (gi − uin)

+ (1 − µ)max {0, λin}max {0, (gi − uin)} (2.53) with µ∈(0, 1] and 1 ≤ i ≤ NC .

We deﬁne the nonlinear complementarity function (NCF) for Tresca friction as

hp hp CT (u , λ ) = max{Fi, kλit + αuitk}λit − Fi(λit + α uit) (2.54) for any positive parameter α > 0 .

Theorem 2.3. The pair (uhp, λhp) satisﬁes the frictional contact conditions (2.51) if and only if

hp hp CT (u , λ ) = 0. (2.55)

The decoupled pointwise non penetration condition is equivalent to

hp hp ϕµ(u |ΓC , λ ) = 0. (2.56)

Proof. For the ﬁrst equivalence see[[41],theoerem5.1]. For the second equivalence we know that the NCF-function satisﬁes

hp hp ϕµ(u |ΓC , λ ) = 0 ⇔ uin ≤ gi, λin ≥ 0, λin(uin − gi) = 0. (2.57)

13 2 hp-BEM For Frictional Contact Problem in Linear Elasticity

Lemma 2.8. The discrete problem (2.34) is equivalent to solving the nonlinear system   Auhp − Dλhp − f 0 = F (uhp, λhp) =  ϕ (uhp| , λhp)  . (2.58)  µ ΓC  hp hp CT (u , λ )

Proof. Auhp − Dλhp − f = 0 is the matrix representation of the ﬁrst equation of hp hp the discrete problem. By Theorem (2.3) the conditions ϕµ(u |ΓC , λ ) = 0 and hp hp CT (u , λ ) = 0 are equivalent to the decoupled pointwise non-penetration condition (2.50) and the frictional condition (2.51), respectively, which are equivalent to (2.34b) by lemma 2.7.

The generalized Newton’s method for the solution of the nonlinear system (2.58) can be deﬁned as

0 (uk, λk) = (uk−1, λk−1) − F (uk−1, λk−1)−1F (uk−1, λk−1), k = 1, 2,... (2.59)

Since F is not diﬀerentiable everywhere, and therefore the Jacobian matrix F 0 (uk−1, λk−1) does not exist everywhere, one has to choose a suitable approximation

0 k−1 k−1 Hk−1 ∈ F (u , λ ) and solve the equation

k k k−1 k−1 −1 k−1 k−1 (u , λ ) = (u , λ ) − Hk−1F (u , λ ), k = 1, 2,... (2.60) which is equivalent to ! δuk H = −F (uk−1, λk−1). (2.61) k−1 δλk

k−1 k−1 Here Hk−1 is the Clarke subdiﬀerential of F at (u , λ ) deﬁned by  A −D  k k k k  ∂ϕµ(u ,λ ) ∂ϕµ(u ,λ )  Hk = ∂u ∂λ . (2.62)  k k k k  ∂CT (u ,λ ) ∂CT (u ,λ ) ∂u ∂λ We obtain (δuk, δλk) by solving the system

 A −D   k−1 k−1  k! F1(u , λ ) k k k k δu  ∂ϕµ(u ,λ ) ∂ϕµ(u ,λ )  = − F (uk−1, λk−1) (2.63)  ∂u ∂λ  k  2  ∂C (uk,λk) ∂C (uk,λk) δλ k−1 k−1 T T F3(u , λ ) ∂u ∂λ where

F1(u, λ) = Au − Dλ − f

F2(u, λ) = ϕµ(u, λ)

F3(u, λ) = CT (u, λ).

14 2.4 Semi-smooth Newton method and algebraic representation

Let S be the set of all nodes on ΓC , and N all remaining nodes . Now, the algebraic representation is given by the linear system

  ANN ANS 0  k−1 k−1   k  F1,N (u , λ ) A A D  δuN k−1 k−1 SN SS S F1,S (u , λ )  k−1 k−1 k−1 k−1   k     ∂ϕµ(u ,λ ) ∂ϕµ(u ,λ )  δu = − . (2.64)  0   S   F (uk−1, λk−1)  ∂uS ∂λS k  2   ∂C (uk−1,λk−1) ∂C (uk−1,λk−1)  δλS k−1 k−1 0 T T F3(u , λ ) ∂uS ∂λS

We remark that the zero block in the coupling matrix D refers to the lines associated with the nodes in set N .

We have

ϕµ(u, λ) = ϕµ(N.u|ΓC , N.λ) = ϕµ(un,S , λn,S ) (2.65)

and

CT (u, λ) = CT (T.u|ΓC , T.λ) = CT (ut,S , λt,S ), (2.66)

where N, T are the algebraic representations of the normal,respectively, tangential vector. Altogether,

  ANN ANS 0  k−1 k−1    F1,N (u , λ ) A A D  δuk  SN SS S  N  k−1 k−1  k−1 k−1 k−1 k−1 k F1,S (u , λ )  ∂ϕµ(u ,λ ) ∂ϕµ(u ,λ )  δu  = −   (2.67)  0 n,S n,S n,S n,S   S   F (uk−1, λk−1)   ∂un,S ∂λn,S  δλk  2   ∂C (uk−1,λk−1) ∂C (uk−1,λk−1)  S k−1 k−1 0 T t,S t,S T t,S t,S F3(u , λ ) ∂ut,S ∂λt,S

15 2 hp-BEM For Frictional Contact Problem in Linear Elasticity with  µ , if λ = u − g = 0  n n  ∂ϕµ µ 1 − √ un−g +(1 − µ)λ , if λ >0 and u >g (u, λ) = 2 2 n n n ∂u λ +(un−g)  µ 1 − √ un−g , otherwise  2 2 λn+(un−g)  µ , if λ = u − g = 0  n n  ∂ϕµ µ 1 − √ λn +(1 − µ)(u − g) , if λ >0 and u >g (u, λ) = 2 2 n n n ∂λ λn+(un−g)  µ 1 − √ λn , otherwise  2 2 λn+(un−g)  −F α , if kλ + αu k ≤ F  t t ∂CT  (u, λ) = α(λ − F) , if kλ + αu k > F ∂u t t t  −α(λt − F) , otherwise  0 , if kλ + αu k ≤ F  t t ∂CT  (u, λ) = λ + kλ + αu k − F , if kλ + αu k > F ∂λ t t t t t  −λt + kλt + αutk − F , otherwise.

n If a function Ψ : R → R is nonnegative and Ψ(x) = 0 if and only if x solves the NCF, then Ψ(x) is called a merit function of the NCF-function. Hence finding a solution of the NCF is equivalent to finding a global minimum of the unconstrained minimization minx∈Rn Ψ(x) with optimal value zero . We define the nonnegative merit function 1 Ψ(u, λ) = F (u, λ)2 (2.68) 2

Algorithm 2.1. (Semi-smooth Newton algorithm)

0 0 n 1. Initialisation: Choose initial solution u , λ ∈ R , ρ > 0, β ∈ (0, 1), σ ∈ 1 (0, 2 ), p > 2 2. For k = 0, 1, 2, ... do

a) Termination Criterion

If k∇Ψ(uk, λk)k < tol or kΨ(uk, λk)k < tol then stop

b) Search Direction Calculation

k k k k k 2n Compute subdiﬀerential Hk ∈ ∂F (u , λ ) and ﬁnd d = (du, dλ) ∈ R s.t

k k k Hkd = −F (u , λ ). (2.69)

16 2.4 Semi-smooth Newton method and algebraic representation

If (2.69) not solvable or if the descent condition

∇Ψ(uk, λk)dk ≤ −ρkdkkp (2.70)

is not satisﬁed, set dk := −∇Ψ(uk, λk).

l Compute search length tk := maxβ : l = 0, 1, 2, ... s.t

k k k k k k k k k Ψ(u + tkdu, λ + tkdλ) ≤ Ψ(u , λ ) + σtk∇Ψ(u , λ )d .

(d) Update

Update the solution vectors and goto step 2.

k+1 k k k+1 k k u = u + tkdu, λ = λ + tkdλ.

Theorem 2.4. The semi-smooth Newton algorithm (2.71) converges locally super- ||ek+1|| linear, i.e. limk→∞ ||ek|| = 0.

T T k+1 k+1 k k −1 k k u , λ = u , λ − Hk F (u , λ ) (2.71)

k k k kT with Hk ∈ ∂F (u , λ ) a subgradient of F at u , λ .

In the frictionless case, i.e. F = 0 and CT (u, λ) := λ, SSN converges locally Q- ||ek+1|| quadratic, i.e. limk→∞ ||ek||2 = const.

Proof. See[8],[10],[53]

Lemma 2.9. (Galerkin orthogonality) Let u ∈ V be an exact solution of the continuous hp problem (2.10) and u ∈ Vhp be the solution of the discrete problem (2.34). There holds

hp hp hp hp hp hSu − Shpu , v iΣ + b(λ − λ , v ) = 0 ∀v ∈ Vhp (2.72)

hp Proof. We choose v ∈ Vhp ⊂ V in (2.10) and subtract (2.10) from the discrete formulation (2.34).

hp Let u ∈ V and u ∈ Vhp. From [15], we deﬁne the following notation 1 ψ := V −1(K + )u 2 1 ψhp := i V −1i∗ (K + )uhp (2.73) hp hp hp 2 1 ψ∗ := V −1(K + )uhp. hp 2

17 2 hp-BEM For Frictional Contact Problem in Linear Elasticity

Lemma 2.10. [15] Let u ∈ V be an exact solution of the continuous problem (2.10) hp and u ∈ Vhp be the solution of the discrete problem. There holds

hp 2 2 hp hp ∗ ku − u kW +kψ − ψhpkV = hSu − Shpu , u − u i + hV (ψh − ψhp), ψ − ψhpi (2.74) where hp hp hp 1 ku − u kW := hW (u − u ), u − u i 2

1 kψ − ψhpkV := hV (ψ − ψhp), ψ − ψhpi 2

˜ 1+ν 1 ν Lemma 2.11. [13, 52, 19] Assume that u ∈ H (Σ) with ν ∈ [0, 2 ] and ψ ∈ H (Γ). There exists a constant C > 0 such that the following approximation properties hold:

1 2 +ν hp h inf ku − v k 1 ≤ C kuk 1+ν hp ˜ 2 H˜ (Σ) v ∈Vhp H (Σ) p 1 +ν h 2 inf kψ − φk 1 ≤ C kψk ˜ ν ˜ − 2 H (Γ) φ∈Whp H (Γ) p

2.5 A priori error estimate

The a priori error estimate is based on the use of the discrete inf-sup condition.

In this section we assume that the discrete inf-sup condition is valid.

Assumption 2.1. There exists a constant βhp > 0 depending on h and p, such that

b(µhp, vhp) βhp ≤ inf sup (2.75) hp hp hp µ ∈Mhp(F) hp kv k 1 kµ k 1 v ∈Vhp ˜ 2 ˜ − 2 H (Σ) H (ΓC )

Lemma 2.12. Let (u, λ) ∈ V × M be the solution of the mixed formulation (2.10)and hp hp (u , λ ) ∈ Vhp × Mhp(F) be the solution of the discrete problem (2.34). Then there hp exists a constant C > 0 independent of h and p such that for all µ ∈ Mhp(F) there holds hp (CS + CEhp ) hp C0 kλ − λ k 1 ≤ C ku − u k 1 + kψ − φk 1 (2.76) ˜ − 2 ˜ 2 ˜ − 2 H (ΓC ) βhp H (Σ) βhp H (Γ) 1 hp +(1 + )kλ − µ k − 1 βhp H 2 (ΓC )

Proof. In this proof we use the notation in[19].

We use the following identity

hp hp hp hp hp Su − Shpu = S(u − u ) + Ehpu = S(u − u ) + Ehp(u − u) + Ehpu

18 2.5 A priori error estimate where

Ehp = S − Shp.

From Lemma 2.5 and Lemma 2.6 Shp and Ehp are continuous. Therefore for all φ ∈ Whp hp hp hp 2 hp 1 1 hSu − Shpu , v i ≤ (CS + CEhp )ku − u k +kEhpuk 1 kv k H˜ 2 (Σ) H− 2 (Σ) H˜ 2 (Σ) hp hp ≤ (CS + CE )ku − u k 1 +C0kψ − φk − 1 kv k 1 . hp H˜ 2 (Σ) H˜ 2 (Γ) H˜ 2 (Σ) (2.77)

Using the Galerkin orthogonality lemma 2.9 and (2.77) we get

hp hp hp hp hp hλ − µ , Vhpi = hSu − Shpu , v i + hλ − µ , Vhpi hp hp ≤ (CS + CE )ku − u k 1 +C0kψ − φk − 1 kv k 1 hp H˜ 2 (Σ) H˜ 2 (Γ) H˜ 2 (Σ) hp hp + kλ − µ k − 1 kv k 1 . (2.78) H 2 (ΓC ) H˜ 2 (Σ)

Using the discrete inf-sup condition (2.75) we obtain hp hp −1 hp kλ − µ k 1 ≤ C(βhp) (CS + CE )ku − u k 1 +C0kψ − φk 1 ˜ − 2 hp ˜ 2 ˜ − 2 H (ΓC ) H (Σ) H (Γ) (2.79) hp +kλ − µ k − 1 H 2 (ΓC )

The triangle inequality and (2.79) yield the assertion.

Theorem 2.5. Let (u, λ) ∈ V × M be the solution of mixed formulation (2.10) such ˜ 1+ν 1 hp hp that u ∈ H (Σ) with ν ∈ [0, 2 ] and (u , λ ) ∈ Vhp × Mhp(F) be the solution of the discrete problem (2.34). We assume that t − Su ∈ L2(Γ) and

kλ k ν +kλ k ν +kFk ≤ kuk ν+1 (2.80) n H (ΓC ) t H (ΓC ) L2(ΓC ) H˜ (Σ) then there exists C > 0 independent of the polynomial degrees p and of the mesh size h such that with (2.73)

h hp hp ku − u k 1 + kψ − ψ k 1 +kλ − λ k 1 ˜ 2 ˜ − 2 ˜ − 2 H (Σ) H (Γ) H (ΓC ) 1 min{ 4 ,ν} −1 h ≤Cβ kuk 1+ν (2.81) hp p H˜ (Σ)

3 We have in particular for u ∈ H˜ 2 (Σ)

h hp −1 1 − 1 ku − u k 1 +kλ − λ k 1 ≤ Cβ h 4 p 4 kuk 3 (2.82) ˜ 2 ˜ − 2 hp ˜ 2 H (Σ) H (ΓC ) H (Σ)

19 2 hp-BEM For Frictional Contact Problem in Linear Elasticity

Proof. We choose

hp hp Khp := {u ∈ Vhp : un (x) ≤ g(x), ∀x ∈ Ghp ∩ ΓC } which is a convex, closed subset of Vhp, and we deﬁne the Lagrange interpolation hp operator Ihp on the set of Gauss-Lobatto points . Let v := Ihpu ∈ Khp. With [[13],Theorems 4.2 and 4.5] we have the following approximation properties: There exist constants C, C1 > 0 independent of u and h, p such that

hp ν −ν 1 ku − v k ≤ Ch p kuk ν , ν > L2(Γ) H (Γ) 2 hp ν−1 1−ν ku − v kH1(Γ) ≤ C1h p kukHν (Γ), ν > 1 (2.83)

From Lemma 2.10 we obtain

hp 2 2 hp hp ∗ hp hp ku − u kW +kψ − ψhpkV = hSu − Shpu , u − u i + hV (ψhp − ψ ), ψ − ψ i.

Since we have

hp hp hp hp hp Su − Shpu = S(u − u ) + Ehpu = S(u − u ) + Ehp(u − u) + Ehpu, we obtain

hp 2 2 hp hp ∗ hp hp ku − u kW +kψ − ψhpkV = hSu − Shpu , u − u i + hV (ψhp − ψ ), ψ − ψ i = hS(u − uhp), u − vhpi + hSuhp, uhp − vhpi + hSu, vhp − uhpi hp hp ∗ hp hp + hEhpu , u − u i + hV (ψhp − ψ ), ψ − ψ i. (2.84)

Using (2.34) and the deﬁnition of Sh we get

hp hp hp hp hp hp hp hp hp hp hp hSu , u − v i = hEhpu , u − v i − b(λ , u − v ) + ht, u − v i (2.85)

Using (2.10), (2.84) and (2.85), we obtain

hp 2 hp 2 hp hp hp hp CW ku − u k 1 +CV kψ − ψ k − 1 ≤ hS(u − u ), u − v i + hSu, v − u i H˜ 2 (Σ) H˜ 2 (Γ) hp hp hp hp hp + hEhpu , u − v i − b(λ , u − v ) + ht, uhp − vhpi − hSu, u − vi − b(λ, u − v) ∗ hp hp + ht, u − vi + hV (ψhp − ψ ), ψ − ψ i hp hp hp hp ≤ hS(u − u ), u − v i + hEhpu , u − v i + ht − Su, uhp − vi + ht − Su, u − vhpi + b(λ, v − uhp) − b(λhp, u − vhp) ∗ hp hp + hV (ψhp − ψ ), ψ − ψ i + hλ − λhp, u − uhpi. (2.86)

20 2.5 A priori error estimate

For every v ∈ K, where CW and CV are the ellipticity constants of V, W.

As in [47], choosing v such that

hp with δu := inf(0, ghp − un ).

Then, we have

hp kv − u kL2(Γ)≤ kg − ghpkL2(ΓC )+kδukL2(ΓC ) (2.88)

hp hp Due to u ∈ Khp we have Ihp inf(g − un , 0) = 0 on ΓC , and with [[13],Theorems 4.2] we have

hp kδu − 0kL2(ΓC ) ≤ kghp − un kL2(ΓC ) 1 −1 1 ≤ C1h p kδu − 0kH (ΓC ) 1 −1 hp 1 ≤ C1h p kghp − un kH (ΓC ).

By interpolation we have

1 − 1 hp 2 2 1 kδu − 0kL2(ΓC )≤ C1h p kghp − un k . (2.89) H 2 (ΓC ) For the ﬁrst term in (2.88) we have

1 − 1 2 2 1 kg − ghpkL2(ΓC )≤ C1h p kgk . (2.90) H 2 (ΓC )

From [47] we know that there exists a constant C2 > 0, independent of h,p, such that

hp ku k 1 ≤ C2kuk 1 . (2.91) H 2 (ΓC ) H 2 (ΓC ) Therefore we have

hp 1 − 1 2 2 1 1 kv − u kL2(Γ)≤ C3h p kgk +kuk . (2.92) H 2 (ΓC ) H 2 (ΓC ) Finally, there exist a constant C such that

hp 1 − 1 2 2 1+ν kv − u kL2(Γ)≤ Ch p kukH (Γ) (2.93)

21 2 hp-BEM For Frictional Contact Problem in Linear Elasticity

1 with ν ∈ [0, 2 ]. For the ﬁrst term in (2.86), employing Cauchy-Schwarz and Young’s inequality ,we obtain

hp hp 1 hp 2 hp 2 hS(u − u ), u − v i ≤ ku − v k 1 + ku − u k 1 2 H˜ 2 (Σ) 2 H˜ 2 (Σ) 2ν+1 1 h 2 hp 2 ≤ 2ν+1 kukH1+ν (Γ)+ ku − u k 1 . 2 p 2 H˜ 2 (Σ) (2.94)

Now we estimate the second term in (2.86).

According to Lemma2.11and Lemma2.6, we get

hp hp hp hp hEhpu , u − v i = hEhp(u − u + u, u − v i

1 hp 2 hp 2 ≤ ku − v k 1 + kEhp(u − u)k − 1 2 H˜ 2 (Σ) 2 H˜ 2 (Σ) 1 2 1 hp 2 + kEhp(u)k − 1 + ku − v k 1 2 H˜ 2 (Σ) 2 H˜ 2 (Σ) 2ν+1 2ν+1 hp 2 h 2 h 2 . ku − u k 1 + 2ν+1 kukH1+ν (Γ)+ 2ν+1 kψk ˜ ν (2.95) H˜ 2 (Σ) p p H (Γ)

We can easily estimate the third term in (2.86). From (2.83) and (2.93), we obtain

hp hp hp hp ht − Su, u − vi + ht − Su, u − v i ≤ kt − SukL2(Σ) ku − vkL2(Σ)+ku − v kL2(Σ) 1 − 1 ≤ Ch 2 p 2 kukH1+ν (Γ). (2.96)

We estimate now the term

hp hp hp b(λ, v − u ) = hλn, vn − un i + hλt, vt − ut i (2.97)

We start with the ﬁrst term in (2.97) ,using (2.93), we get Z hp hp hp hλn, vn − un i = λn(vn − un ) ds ≤ kλnkL2(ΓC )kv − u kL2(Γ) ΓC 1 1 2 − 2 2 ≤ Ch p kukH1+ν (Γ) (2.98)

Due to the approximation (2.87) the second term in (2.97) is

hp hλt, vt − ut i = 0. (2.99)

We estimate the term

hp hp hp hp hp hp b(λ , u − v ) = hλn , un − vn i + hλt , ut − vt i (2.100)

22 2.5 A priori error estimate

Employing Cauchy Schwarz and Young’s inequality, we have Z Z hp hp hp hp hp hλn , un − vn i = (λn − λn)(un − vn ) ds + λn(un − vn ) ds ΓC ΓC hp hp hp 1 1 ≤ kλn − λnk − kun − vn k +kλnkL2(ΓC )kun − vn kL2(ΓC ) H 2 (ΓC ) H 2 (ΓC ) 1 h2ν+1 hν+1 ≤ kλhp − λ k2 + kuk2 + kuk2 n n − 1 2ν+1 H1+ν (Γ) ν+1 H1+ν (Γ) 2 H 2 (ΓC ) 2 p p (2.101) Similarly to (2.101), we get 1 h2ν+1 hν+1 hλhp, u − vhpi ≤ kλhp − λ k2 + kuk2 + kuk2 (2.102) t t t t t − 1 2ν+1 H1+ν (Γ) ν+1 H1+ν (Γ) 2 H 2 (ΓC ) 2 p p Combining (2.101) and (3.57), we obtain 2ν+1 ν+1 hp hp hp 1 h 2 h 2 b(λ , u − v ) ≤ kλ − λ k 1 + kuk 1+ν + kuk 1+ν (2.103) ˜ − 2 2ν+1 H (Γ) ν+1 H (Γ) 2 H (ΓC ) 2 p p and we have, see[19]

∗ hp hp 1 hp hV (ψhp − ψ ), ψ − ψ i ≤ (CK + )ku − u k 1 kψ − φk − 1 2 H˜ 2 (Σ) H˜ 2 (Γ) hp + CV kψ − ψ k − 1 kψ − φk − 1 H˜ 2 (Γ) H˜ 2 (Γ) 1 1 2 2 hp 2 ≤ (CK + ) kψ − φk − 1 + ku − u k 1 2 2 H˜ 2 (Γ) 2 H˜ 2 (Σ) 1 2 2 hp 2 + CV kψ − φk − 1 + kψ − ψ k − 1 ∀φ ∈ Whp 2 H˜ 2 (Γ) 2 H˜ 2 (Γ) (2.104) Finally we estimate the term

hp h hp hp hp hp hλ − λ , u − u i = hλn − λn , un − un i + hλt − λt , ut − ut i. (2.105)

hp hp Choosing µt = λt and µn = 0, µn = 2λn in the inequality (2.10b) and µt = λt and hp hp hp µn = 0, µn = 2λn in the discrete inequality (2.34b), we obtain the complementary conditions Z Z hp hp λn(un − g) ds = λn (un − g) ds = 0 (2.106) ΓC ΓC

Let πMhp be the L2-projection operator mapping Mhp deﬁned by ( πMn,hp : L2(ΓC ) −→ Mn,hp πMhp = (2.107) πMt,hp : L2(ΓC ) −→ Mt,hp which satisﬁes Z Z

(λn − πMn,hp λ)µn ds = 0, (λt − πMt,hp λ)µt = 0 ds ∀µ = (µn, µt) ∈ Mhp ΓC ΓC (2.108)

23 2 hp-BEM For Frictional Contact Problem in Linear Elasticity and

hν kλ − π λk ≤ C kλ k ν (2.109) n Mn,hp L2(ΓC ) p n H (ΓC )

hν kλ − π λk ≤ C kλ k ν (2.110) t Mt,hp L2(ΓC ) p t H (ΓC ) for any real number ν ≥ 0 .

We obtain

hπMn,hp λ − λn, vn − πMt,hp vi 1 kπMn,hp λ − λnk − = sup H 2 (ΓC ) 1 kvnk 1 06=vn∈H 2 (ΓC ) H 2 (ΓC )

kπM λ − λnk kvn − πM vk ≤ sup n,hp L2(ΓC ) n,hp L2(ΓC ) 1 kvnk 1 06=vn∈H 2 (ΓC ) H 2 (ΓC ) 1 h 2 ≤ C kπ λ − λ k (2.111) p Mn,hp n L2(ΓC ) and

1 h 2 1 kπMt,hp λ − λtk − ≤ C kπMt,hp λ − λtkL2(ΓC ) (2.112) H 2 (ΓC ) p

According to the continuity of the Direchlet-to-Neumann operator, we have

kλnk 1 +kλtk 1 ≤ Ckuk 3 . (2.113) H 2 (ΓC ) H 2 (ΓC ) H˜ 2 (Σ)

hp hp We choose µn = πMn,hp λ and µt = πMt,hp λ, as a consequence we get

1 2 +ν hp h inf kλn − µ k 1 ≤ C kuk ν+1 n − 2 ˜ µn∈Mn H (ΓC ) p H (Σ) 1 +ν hp h 2 inf kλt − µ k 1 ≤ C kuk ν+1 (2.114) t − 2 H˜ (Σ) µt∈Mt(F) H (ΓC ) p

We start with the ﬁrst term in (2.105). According to (2.106), we have

Z Z hp hp hp hp hλn − λn , un − uni = λn(un − g) ds + λn (un − g) ds (2.115) ΓC ΓC

The normal component can be written as

hp X hp X hp X un := αinφi, µn := βinψi, λn := λinψi. (2.116) i∈NC i∈NC i∈NC

24 2.5 A priori error estimate

hp For µn ∈ Mn,hp we have

Z hp hp X hp µn (un − g) ds = αinµinDi − hµn , gi ΓC i X X Z = αinµinDi − µin gψi ds i i ΓC X X = αinµinDi − µingiDi ds i i X = (αin − gi)µinDi ≤ 0 (2.117) i

hp Choosing µn = πMn,hp λn, from (2.117) we get

Z Z Z hp hp hp hp hp λn(un − g) ds = (λn − µn )(un − g) ds + µn (un − g) ds ΓC ΓC ΓC Z hp hp ≤ (λn − µn )(un − g) ds ΓC Z hp ≤ (λn − πMn,hp λn)(un − g) ds ΓC Z hp = (λn − πMn,hp λn)((un − g) − (un − g)) ds ΓC Z

+ (λn − πMn,hp λn)(un − g − πMn,hp (un − g)) ds ΓC Z hp = (λn − πMn,hp λn)(un − un) ds ΓC Z

+ (λn − πMn,hp λn)(un − πMn,hp un + πMn,hp g − g) ds ΓC (2.118)

25 2 hp-BEM For Frictional Contact Problem in Linear Elasticity and we obtain

Z hp hp 1 1 λn(un − g) ds ≤ kπMn,hp λn − λnk − kun − unk H 2 (ΓC ) H 2 (ΓC ) ΓC

+ kπMn,hp λn − λnkL2(ΓC )kun − πMn,hp unkL2(ΓC )

+ kπMn,hp λn − λnkL2(ΓC )kg − πMn,hp gkL2(ΓC )

1 2 +ν h hp ≤ C kλnkHν (Γ )ku − u k 1 p C H˜ 2 (Σ) ! h2ν h2ν + kλ k ν kuk 1+ν + kλ k ν kgk ν p n H (ΓC ) H˜ (Σ) p n H (ΓC ) H (ΓC ) 2ν hp 2 h 2 ≤ ku − u k 1 +C kuk ˜ 1+ν . H˜ 2 (Σ) p H (Σ) (2.119)

We consider the hp-Lagrange interpolation operator Ihp deﬁned on the Gauss-Lobatto points mapping onto Vhp.

The linear combination of Ihpun can be written as

X Ihpun := ainφi with ain ≤ gi (2.120) i where

1 Z Z gi := gψi ds, Di = φi ds > 0 (2.121) Di ΓC ΓC

Using the linear combination in (2.116) and the biorthogonality condition, we get

Z hp X hp λn (Ihpun − g) ds = ainλinDi − hλn , gi ΓC i X X Z = ainλinDi − λin gψi ds i i ΓC X X = apnλinDi − λingiDi ds i i X = (ain − gi)λinDi ≤ 0. (2.122) i

26 2.5 A priori error estimate

According to (2.122)

Z Z Z hp hp hp λn (un − g) ds = λn ((un − g) − (Ihpun − g)) ds + λn (Ihpun − g) ds ΓC ΓC ΓC Z hp ≤ λn (un − Ihpun) ds ΓC Z Z hp = (λn − λn)(un − Ihpun) ds + λn(un − Ihpun) ds ΓC ΓC

hp ≤ kλn − λnk − 1 kun − Ihpunk 1 H 2 (ΓC ) H 2 (ΓC )

+ kλnkL2(ΓC )kun − IhpunkL2(ΓC )

hν+1 ≤ kλhp − λ k2 +C kuk2 (2.123) n n − 1 ν+1 ˜ ν+1 H 2 (ΓC ) p H (Σ)

We now estimate the term the second term in (2.105)

From (2.34b), we have

hp hp hp hp hp λt ∈ Mt,hp, hµt − λt , ut i ≤ 0, ∀µt ∈ Mt,hp (2.124)

hp Choosing µt = πMt,hp λ, from (2.124) we get

hp hp hp hp hp hp hp hλt − λt , ut − uti = hλt − µt , ut − uti + hµt − λt , ut − uti hp hp hp hp ≤ hλt − µt , ut − uti − hµt − λt , uti hp hp hp hp ≤ hλt − µt , ut − uti + hλt − µt , uti + hλt − λt, uti. (2.125)

The estimate of the ﬁrst term in (2.125) gives

Z hp hp hp hp hλt − µt , ut − uti = (λt − µt )(ut − ut) ds ΓC 1 hp 2 hp 2 ≤ kλt − µt k − 1 + kut − utk 1 2 H 2 (ΓC ) 2 H 2 (ΓC ) 1 +ν h 2 2 hp 2 ≤ C 1 kuk ν+1 + kut − utk 1 . +ν H˜ (Σ) 2 p 2 2 H (ΓC ) (2.126)

27 2 hp-BEM For Frictional Contact Problem in Linear Elasticity

We now estimate the second term in (2.125) Z hp hp hλt − µt , uti = (λt − µt )ut ds ΓC hp ≤ kλt − µt k − 1 kutk 1 H 2 (ΓC ) H 2 (ΓC ) 1 +ν h 2 2 ≤ C kuk ν+1 . 1 +ν H˜ (Σ) p 2 (2.127)

hp The linear combination of Ihput and λt can be written as X hp X Ihput := aitφi, λt := λitψi. (2.128) i∈NC i∈NC Using the linear combination in (2.128), the biorthogonality condition and the fact that

|λit| ≤ F, we get Z hp X X λt Ihput ds = aitλitDi ≤ F aitDi ΓC i i X Z ≤ F ait φi ds i ΓC Z ≤ F |Ihput| ds. (2.129) ΓC

1 + kλtkL2(ΓC )kut − Ihputk H 2 (ΓC )

+ kFkL2(ΓC )kut − IhputkL2(ΓC ) ν+ 1 hp 2 h 2 2 ≤ kλt − λtk 1 + 1 kuk ν+1 − 2 ν+ H˜ (Σ) H (ΓC ) p 2 (2.130)

28 2.6 A posteriori error estimates for contact with friction

Now combining (2.126) ,(2.127),(2.130) we get the estimate of the second term in(2.105)

ν+ 1 hp hp hp 2 hp 2 h 2 2 hλt − λt , ut − uti ≤ kλt − λtk 1 +kut − utk 1 + 1 kuk ν+1 − 2 2 ν+ H˜ (Σ) H (ΓC ) H (ΓC ) p 2 (2.131) Finally, the estimate follows immediately by using Lemma2.12, Lemma(2.11), (2.131) , (2.94), (2.95), (2.96), (2.104), (2.98), (3.58), (2.119), (2.123).

2.6 A posteriori error estimates for contact with friction

Let (u, λ) be the exact solution of the continuous problem (2.10) and let (uhp, λhp) be the solution of the discrete boundary element problem (2.34). We now give a computable upper bound for |ku − uhpk|2, where

hp 2 hp 2 hp 2 hp |ku − u k| := ku − u k 1 +kψ − ψ k 1 +kλ − λ k 1 (2.132) − ˜ − 2 H˜ 2 (Σ) H˜ 2 (Γ) H (ΓC ) Theorem 2.6. Let (u, λ) be the exact solution of the boundary problem (2.10) and (uhp, λhp) be the solution of the discrete boundary problem (2.34). Then there holds the estimate :

hp 2 2 X 2 hp + hp + ku − u k 1 +kψ − ψhpk − 1 . ηhp(I) + h(λn ) , (g − un ) iΓC H˜ 2 (Σ) H˜ 2 (Γ) I∈Thp

hp − 2 1 hp + 2 + k(λn ) k − 1 + k(un − g) k 1 H˜ 2 (ΓC ) 4 H 2 (ΓC ) hp 2 hp + 2 + kλ − λ k 1 +k(|λt | − F) k 1 ˜ − 2 ˜ − 2 H (ΓC ) H (ΓC ) Z hp − hp hp hp − + |(|λt | − F) ||ut | + 2(λt ut ) ds, ΓC (2.133) where h h η2 (I) = I kt − S uhpk2 + I k(−λhp) − S uhpk2 hp hp L2(I∩ΓN ) hp L2(I∩ΓC ) pI pI ∂ ∗ hp 2 + hI k (V (ψ − ψ ))k ∂s hp L2(I)

Proof. Using Lemma 2.10, since hW ·, ·i, hV ·, ·i are positive deﬁnite, there exist constants CW ,CV > 0 such that

hp 2 hp hp hp CW ku − u k 1 +CV kψ − ψ k − 1 ≤ hSu − Shpu , u − u i H˜ 2 (Σ) H˜ 2 (Γ) ∗ hp hp + hV (ψhp − ψ ), ψ − ψ i

29 2 hp-BEM For Frictional Contact Problem in Linear Elasticity

Using Galerkin orthogonality, we obtain

hp 2 hp hp hp CW ku − u k 1 + CV kψ − ψ k 1 ≤ hSu − Shpu , u − u i H˜ 2 (Σ) H˜ 2 (Γ) hp hp hp hp hp hp + hSu − Shpu , u − v iΣ + b(λ − λ , u − v ) ∗ hp hp + hV (ψhp − ψ ), ψ − ψ i hp hp ∗ hp hp ≤ hSu − Shpu , u − v i + hV (ψhp − ψ ), ψ − ψ i + b(λ − λhp, uhp − vhp) hp hp hp hp ≤ ht, u − v iΓN − b(λ, u − v ) − hShpu , u − v i ∗ hp hp hp hp hp + hV (ψhp − ψ ), ψ − ψ i + b(λ − λ , u − v ) hp hp hp hp hp ≤ ht − Shpu , u − v iΓN + h(−λ ) − Shpu , u − v iΓC hp hp ∗ hp hp + b(λ − λ , u − u) + hV (ψhp − ψ ), ψ − ψ i ≤ A + B + C + D (2.134)

We estimate the terms A and B, employing the Cauchy-Schwarz inequality. We obtain:

X hp hp hp A + B ≤ k(−λ ) − Shpu kL2(I)ku − v kL2(I) I∈Thp∩ΓC X hp hp + kt − Shpu kL2(I)ku − v kL2(I) (2.135) I∈Thp∩ΓN

Let Πhp be the hp-Cl´ement interpolation operator mapping onto Vhp [48]. Then there holds

1 h 2 ku − ΠhpukL (I) ≤ C kuk 1 , (2.136) 2 p H˜ 2 (ω(I)) where ω(I) consists of all neighboring elements of I.

hp hp hp We choose in (2.135) v = u + Πhp(u − u ) we obtain the following approximation:

1 2 hp hI hp ku − v kL2(I)≤ C ku − u k 1 (2.137) pI H 2 (ω(I))

We apply a result from [[15],Theorem 5.1] for the term D, to obtain

∗ hp hp ∗ hp hp D := hV (ψ − ψ ), ψ − ψ i ≤ kV (ψ − ψ )k 1 kψ − ψk 1 hp hp H 2 (Γ) H− 2 (Γ) 1   2 X ∂ ∗ hp 2 hp ≤ c  hI k (V (ψhp − ψ ))k  kψ − ψk − 1 (2.138) ∂s L2(I) H 2 (Γ) I∈Thp

Finally we estimate the term C. In order to obtain an a posteriori error estimate, we have to estimate the term:

hp hp hp hp hp hp C = b(λ − λ , u − u) = hλn − λn , un − uni + hλt − λt , ut − uti. (2.139)

30 2.6 A posteriori error estimates for contact with friction

hp + + Using the condition hλn, un − gi = 0 and h(λn ) , un − gi ≤ 0 where v = max{0, v} and v− = min{0, v}, i.e. v = v+ + v−.

hp hp hp + hp hp − hp hλn − λn , un − uni = hλn − (λn ) , un − uni − h(λn ) , un − uni hp + hp hp − hp = hλn − (λn ) , un − g + g − uni − h(λn ) , un − uni hp + hp = hλn − (λn ) , un − gi + hλn, g − uni hp + hp − hp − h(λn ) , g − uni − h(λn ) , un − uni hp + hp hp − hp ≤ hλn − (λn ) , un − gi − h(λn ) , un − uni hp + hp hp + hp − = h(λn ) , g − un i − hλn, (g − un ) + (g − un ) i hp − hp − h(λn ) , un − uni hp + hp hp hp + hp − hp − ≤ h(λn ) , g − un i + hλn − λn − (λn ) − (λn ) , (g − un ) i hp − hp − h(λn ) , un − uni hp + hp + hp hp − = h(λn ) , (g − un ) i + hλn − λn, (g − un ) i hp − hp − hp − hp − h(λn ) , (g − un ) i − h(λn ) , un − uni hp + hp + hp hp − = h(λn ) , (g − un ) i + hλn − λn, (g − un ) i hp − hp − h(λn ) , un − uni hp − hp ≤ k(λn ) k − 1 kun − unk 1 H˜ 2 (ΓC ) H˜ 2 (Σ) hp hp − hp + hp + + kλn − λnk − 1 k(g − un ) k 1 +h(λn ) , (g − un ) iΓC . H˜ 2 (ΓC ) H 2 (ΓC ) (2.140)

Using the condion −λtut + F|ut| = 0 and λt = ξF with |ξ| ≤ 1:

hp hp hp hp hp hp hλt − λt , ut − uti = −λtut + λt ut + λtut − λt ut hp hp hp hp = −F|ut| + λt ut + ξFut − λt ut hp hp hp hp ≤ −F|ut| + λt ut + F|ut | − λt ut hp + hp hp hp ≤ (|λt | − F) |ut| + F|ut | − λt ut hp + hp hp + hp hp hp hp ≤ (|λt | − F) |ut − ut | + (|λt | − F) |ut | + F|ut | − λt ut hp + hp ≤ k(|λt | − F) k − 1 kut − ut k 1 H˜ 2 (ΓC ) H˜ 2 (ΓC ) hp + hp hp hp hp hp hp + [(|λt | − F) − (|λt | − F)]|ut | − λt ut + |λt ||ut | hp + hp ≤ k(|λt | − F) k − 1 kut − ut k 1 H˜ 2 (ΓC ) H˜ 2 (ΓC ) hp − hp hp hp − + |(|λt | − F) ||ut | + 2(λt ut ) (2.141)

hp Using Young’s inequality we throw the term ku − u k 1 to the left hand side, and H˜ 2 (Σ) we obtain the estimate of the theorem.

Now we look for an upper bound of the discretization error λ − λhp.

31 2 hp-BEM For Frictional Contact Problem in Linear Elasticity

Lemma 2.13. Let (u, λ) solve the saddle point problem (2.10), and let (uhp, λhp) be the solution of the discrete problem (2.34). Then there holds hp 2 0 hp 2 hp 2 00 X 2 kλ − λ k − 1 ≤ C ku − u k 1 +kψ − ψ k − 1 + C ξhp(I) H 2 (ΓC ) H˜ 2 (Σ) H˜ 2 (Γ) I⊂Thp (2.142) Where h h ξ2 (I) = I kt − S uhpk2 + I k(−λhp) − S uhpk2 (2.143) hp hp L2(I∩ΓN ) hp L2(I∩ΓC ) pI pI

hp Proof. For v ∈ V and v ∈ Vhp we have

hλ − λhp, vi = hλ − λhp, v − vhpi + hλ − λhp, vhpi

Using Galerkin orthogonality and the formulation (2.10a), we obtain

hp hp hp hp hp hλ − λ , vi = hλ − λ , v − v i − hSu − Shpu , v iΣ = L(v − vhp) − hSu, v − vhpi − hλhp, v − vhpi hp hp hp − hSu − Shpu , viΣ − hSu − Shpu , v − viΣ hp hp hp hp hp = ht − Shpu , v − v iΓN + h(−λ ) − Shpu , v − v iΓC hp − hSu − Shpu , viΣ = A + B + C (2.144)

hp hp X hp hp A = ht − Shpu , v − v iΓN ≤ kt − Shpu kL2(I)ku − v kL2(I) (2.145) I∈Thp∩ΓN hp hp hp X hp hp hp B = h(−λ ) − Shpu , v − v iΓC ≤ k(−λ ) − Shpu kL2(I)ku − v kL2(I) I∈Thp∩ΓC (2.146)

hp hp C = hSu − Shpu , viΣ ≤ (Cs + CE )ku − u k 1 kvk 1 hp H˜ 2 (Σ) H˜ 2 (Σ) hp + C0kψ − ψ k − 1 kvk 1 (2.147) H˜ 2 (Γ) H˜ 2 (Σ)

hp We choose v = Πhpv in A and B, we get hp hp hλ − λ , vi ≤ (Cs + CE )ku − u k 1 +C0kψ − ψhpk − 1 kvk 1 hp H˜ 2 (Σ) H˜ 2 (Γ) H˜ 2 (Σ)

 1  2 X hI hp 1 + C  kt − Shpu kL2(I) kvk pI H˜ 2 (Σ) I∈Thp∩ΓN

 1  2 X hI hp hp 1 + C  k(−λ ) − Shpu kL2(I) kvk pI H˜ 2 (Σ) I∈Th∩ΓN (2.148)

32 2.7 Numerical Experiments

Using the deﬁnition of the dual norm and (a+b)2 ≤ 2a2 +2b2, the assertion immediately follows.

Theorem 2.7. Let (u, λ) be the exact solution of the boundary problem (2.10) and (uhp, λhp) be the solution of the discrete boundary problem (2.34), then there holds the a posteriori estimate :

hp 2 hp 2 X 2 hp + hp + ku − u k 1 +kλ − λ k 1 ηhp(I) + h(λn ) , (g − un ) iΓC ˜ 2 ˜ − 2 . H (Σ) H (ΓC ) I∈Thp hp − 2 hp + 2 + k(λn ) k − 1 +k(un − g) k 1 H˜ 2 (ΓC ) H 2 (ΓC ) hp + 2 + k(|λt | − F) k − 1 H˜ 2 (ΓC ) Z hp − h hp hp − + |(|λt | − F) ||ut | + 2(λt ut ) ds ΓC (2.149) where h h η2 (I) = I kt − S uhpk2 + I k(−λhp) − S uhpk2 hp hp L2(I∩ΓN ) hp L2(I∩ΓC ) pI pI ∂ ∗ hp 2 + hI k (V (ψ − ψ ))k (2.150) ∂s hp L2(I)

Proof. Follows immediately from Lemma 2.13 and Theorem 2.6.

2.7 Numerical Experiments

Numerical results are presented with the MATLAB package of L.Banz for the contact of the two-dimentional elastic body Ω = [−0.5, 0.5]2 with a rigid straight line. The contact boundary ΓC = [−0.5, 0.5] × −0.5 comes in contact with rigid obstacle which occupies the half space y ≤ −0.5. The Young’s modulus and the Poisson’s ratio are E = 200, ν = 0.3 respectively. The gap is assumed to be zero, i.e g = 0 and the given friction function F = 0.3.

The applied Neumann boundary forces on the top, the left and the right side of the domain are give by see [42]

1 1 3 2 ! −400 sign(x)(y + 2 )( 2 − y)exp(−10(y + 10 ) ) tside = 1 1 10(y + 2 )( 2 − y) ! 0 ttop = 1 2 1 2 −500( 2 − x) ( 2 + x)

In our numerical experiments we use the semi-smooth Newton Algorithm 2.1 to solve the discrete problem. The initial mesh is uniform and consists of 16 elements. We

33 2 hp-BEM For Frictional Contact Problem in Linear Elasticity introduce an hp-adaptive algorithm based on locally testing for smoothness as done in [40] to decide of whether to perform h-or-p-reﬁnement. The details of the procedure are described in Algorithm 2.2 .

Algorithm 2.2. (Mesh reﬁnement strategy)

1. Generate an initial mesh Thp,0, discrete spaces Vhp,0, Whp,0, set l = 0

2. Choose a tolerence T OL > 0 and steering parameter 0 ≤ γ ≤ 1

3. For l = 0, 1, 2, ...

hp hp a) Solve the discrete problem, for (ul , λl )

based on the hp-mesh Thp,l

b) Compute indicators ηI for all segments I ∈ Thp,l 1 and the global error estimator η = P η2 2 I∈Thp,l I

1 c) Stop if P η2 2 ≤ TOL I∈Thp,l I

d) Compute the local approximation ΞI of ηI and the global approximation P Ξ2 of η I∈Thp,l I e) For (θ ∈ (0, 1)), mark all elements in N         ˆ X 2 X 2 N = argmin N ⊂ Thp,l : Ξ ≥ θ Ξ I I   I∈Nˆ I∈Thp,l  

f) Estimate analyticity [40] (δ ∈ (0, 1))

i. Compute Legendre coeﬃcients of uhp|I with

p XI 2i + 1 Z u (x)| = a L (x), a = u (x)L (x) hp I i i i 2 hp i i=0 I

ii. Compute the slope mI ∈ R and bI ∈ R, with

X 2 mI , bI ∈ R : (i · mI + bI − |log |ai||) → min i

iii. if exp(−mI ) ≤ δ increase pI by one, else bisect I ∈ N and keep the polynomial degree equal to pI on the resulting sub-elements.

g) Compute the new hp-mesh Thp,l+1

34 2.7 Numerical Experiments

h) Generate the discrete spaces , Vhp,l+1, Whp,l+1 based on the mesh Thp,l+1

i) Set l = l + 1 , go to (a)

In Figure2.1 we show the deformed configuration. Figure 2.2 shows the estimated errors for the h-uniform, h-adaptive and hp-adaptive methods with (θ = 0.5, δ = 0.6). Figure2.3 shows the normal and the tangential component of the Lagrange multiplier. The refined meshes and polynomial degrees obtained with our approach are shown in Figure 2.4 and Figure 2.5. The first figure shows the adaptivety generated meshes and polynomial degrees after 11 and 13 refinement steps using ηbub (the bubble indicator) as error indicator see [47]. The second figure shows adaptivety refined meshes and polynomial degrees obtained after 9 and 11 steps using the residual indicator .

35 2 hp-BEM For Frictional Contact Problem in Linear Elasticity

Figure 2.1: Deformed geometry

2 10 unf. h, p=1, energy diff. unf. h, p=1, bubble unf. h, p=1, residual 1 10 residual h−adap, p=1, bubble residual h−adap, p=1, residual bubble hp−adap, bubble residual hp−adap, residual 0 10

−1 10 Error

−2 10

−3 10

−4

10 1 2 3 4 10 10 10 10 Degrees of Freedom

Figure 2.2: Estimated errors

36 2.7 Numerical Experiments

−2

−4 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

(a) normal component λn

0.8

0.6

0.4

0.2

−0.2

−0.4

−0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

(b) tangential component λt

Figure 2.3: Visualization of Lagrange multiplier

37 2 hp-BEM For Frictional Contact Problem in Linear Elasticity

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 111111111 1 1 1 1 111111 1 1 1 1 111111111

(a) h-adaptivity, θ = 0.5, δ = 0.6

1 1 1 2 2 2 1 1 2 2 2 1 1 1 1 1 1 1

2 2

1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 3 3

3 3

4 4 11111111 2 2 1 11111111 1 2 1 1 1 11111111

(b) hp-adaptivity, θ = 0.5, δ = 0.6 38 Figure 2.4: Adaptivety generated meshes for Lam´e-BEM(bubble indicator) 2.7 Numerical Experiments

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11111111111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11111111111

(a) h-adaptivity, θ = 0.5,δ = 0.6

11 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 11 1 1 2 2

3 3

1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 1 1 2 2 2 2 3 3

3 3

2 4 2 2111222 3 2 2 1 1 1 1 1 2 2 1 2 2 2 2 1 1 2221112

(b) hp-adaptivity, θ = 0.5,δ = 0.6 39 Figure 2.5: Adaptivety generated meshes for Lam´e-BEM(residual indicator)

3 Stabilized mixed hp-BEM in Linear Elasticity

In this chapter, we consider a contact problem in 2D elasticity with Tresca friction. We consider a mixed boundary integral formulation, which is stabilized following ideas of P.Hild, Y. Renard and V.Lleras [36],[37],[45] for the FEM. Here a mesh-dependent stabilization term is added to the discrete mixed formulation, in order to avoid the discrete inf-sup condition.. First we study the existence and uniqueness of the solution of the discrete problem. A subsection is devoted to a priori error and a posteriori error estimates. Finally, we present some numerical experiments, which compare the stabilized and the non-stabilized cases.

3.1 The mixed formulation

2 The notations in Chapter 2 are used in this chapter. Let Ω ⊂ R be a bounded Lipschitz domain with the boundary Γ := ∂Ω = ΓN ∪ ΓD ∪ ΓC be decomposed into the non-intersecting Neumann conditioned segment ΓN , the Dirichlet conditioned segment ΓD and the contact conditioned segment ΓC which potentially can come in contact with the rigid foundation with Γ¯C ∩ Γ¯D = ∅ for simplicity. The problem then consists in ﬁnding the displacement ﬁeld u :Ω −→ R such that − div σ(u) = 0 in Ω (3.1a) σ(u) = C : (u) in Ω (3.1b)

u = 0 on ΓD (3.1c)

σ(u) · n = t on ΓN (3.1d)

σn ≤ 0, un ≤ g, σn(un − g) = 0 on ΓC (3.1e)

|σt| ≤ F, σtut + F|ut| = 0 on ΓC (3.1f)

Recall that the scalar normal and tangential boundary stresses are deﬁned as

σn := n · σ(u) · n and σt := t · σ(u) · n, and for any displacement u and for surface stresses deﬁned on ∂Ω we adopt the following notation

u = unn + utt and σ(u)n = σn(u)n + σt(u)t (3.2)

41 3 Stabilized mixed hp-BEM in Linear Elasticity

As in Chapter2, we introduce the function spaces

1 1 d 1 V := [H˜ 2 (Σ)] = H˜ 2 (Σ) := {u ∈ H 2 (Σ); supp(u) ⊂ Σ}

1 2 V := {u ∈ H˜ (Σ) : u = 0 on ΓD} 1 d 1 W := [H 2 (ΓC )] = H 2 (ΓC ) − 1 d − 1 M := [H 2 (ΓC )] = H 2 (ΓC ) where

Σ¯ := Γ¯C ∪ Γ¯N 2 − 1 1 and we assume that F ∈ L (ΓC ), t ∈ H 2 (ΓN ) and g ∈ H 2 (ΓC ).

We deﬁne the DtN (Dirichlet-to-Neumann) mapping u|Γ → σ(u)· n. There holds

σn(u) ≡ Su · n|ΓC , σt(u) ≡ Su · t|ΓC . (3.3)

The Steklov-Poincar´eoperator S is deﬁned by 1 1 S := W + (K0 + )V −1(K + ). 2 2

We deﬁne the space for the Lagrange multiplier λ by

M = Mn × Mt where

1 1 ˜ − 2 2 Mn := {µn ∈ H (ΓC ): hµn, vniΓC ≤ 0, ∀v ∈ H (ΓC ) with vn ≤ 0 a.e on ΓC } and

Mt(F) := {µt ∈ L2(ΓC ): |µt|≤ F a.e on ΓC }. are the sets of normal and tangential Lagrange multipliers.

The classical formulation (3.1) can be rewritten in a weak sense as a saddle point problem as follows:

Find (u, λ) ∈ V × M such that

hSu, viΣ + b(λ, v) = ht, viΓN ∀v ∈ V (3.4a) b(µ − λ, u) hg, µ − λ i ∀ µ ∈ M (3.4b) 6 n n ΓC with the functional

b(µ, v) := hµ , v i + hµ , v i . (3.5) n n ΓC t t ΓC

1 − 1 Here the notation h·, ·i represents the duality pairing between H 2 (ΓC ) and H˜ 2 (ΓC ).

42 3.2 The stabilized mixed hp-BEM formulation

3.2 The stabilized mixed hp-BEM formulation

Let Thp be a subdivision of Γ¯C ∪ Γ¯N into straight line segments I. We associate each element of Thp with a polynomial degree pI ≥ 1 and set p = (pI )I∈Thp . Furthermore we deﬁne the set of Gauss-Lobatto points GI,hp on each element I ∈ Thp of corresponding polynomial degree pI as the aﬃne mapping of the Gauss-Lobatto points on [−1, 1] onto

I and set Ghp := ∪I∈Thp GI,hp.. We introduce the space of continuous piecewise polynomials for the discretization of u:

1 hp 0 hp 2 hp 2 Vhp := {u ∈ C (Σ) : ∀I ∈ Thp, u |I ∈ [PpI (I)] , u = 0 on ΓD} ⊂ H (Γ) and the space of piecewise polynomials for the discrete tractions

1 2 − 2 Whp := {L2(Σ) : ∀I ∈ Th, φ|I ∈ [PpI −1(I)] } ⊂ H (Γ).

An explicit representation of V −1 is not known, which causes additional diﬃculties in the numerical treatment. To resolve this problem we need to approximate the Steklov- Poincar´eoperators.

− 1 1 Let ihp : Whp ,→ H 2 (Γ) and jhp : Vhp ,→ H 2 (Γ) denote the canonical imbeddings ∗ ∗ with dual maps ihp and jhp.

The approximation Shp of the Poincar´e-Steklov operators is given by 1 1 S := W + (K0 + )i (i∗ V i )−1i∗ (K + ) (3.6) hp 2 hp hp hp hp 2

Recall that the operator Ehp = S − Shp represents the error in the approximation of the Steklov-Poincar´eoperator (see Chapter2).

Let THq denote an additional partition of ΓC , which needs not to coincide with Thp|ΓC . We deﬁne the discrete version of the space M for the Lagrange multiplier as (cf [19])

MHq(F) := Mn,Hq × Mt,Hq(F), where

Hq Hq Mn,Hq := {λn ∈ WHq : λn (x) ≥ 0 ∀x ∈ ΓC ∩ GHq}

Hq Hq Mt,Hq := {λt ∈ WHq : |λt (x)| ≤ F ∀x ∈ ΓC ∩ GHq} and Hq Hq WHq := {λ ∈ L2(ΓC ): ∀J ∈ THq, λ |J ∈ Pq(J)}

43 3 Stabilized mixed hp-BEM in Linear Elasticity

Remark 3.1. It follows from (3.4) that λ = −σ(u) · n in a weak sense. Therefore, λn has an interpretation as the negative normal contact traction. We consider approxima- Hq h hp Hq h hp tions λn and −σn(u )(resp. λt and −σt (u )) of λn = −σn(u) (resp.λt = −σt(u)). where h hp hp h hp hp σn(u ) := Shpu · n|ΓC , σt (u ) := Shpu · t|ΓC . (3.7) The discretized version of (3.4) with stabilized Lagrange multiplier reads as:

hp Hq Hq Hq Find u ∈ Vhp and λ = (λn , λt ) ∈ MHq(F) := Mn,Hq × Mt,Hq(F): Z hp hp Hq hp Hq h hp h hp hShpu , v iΣ + b(λ , v ) + γ(−λn − σn(u ))σn(v ) ds ΓC Z Hq h hp h hp hp hp + γ(−λt − σt (u ))σt (v ) ds = L(v ) ∀v ∈ Vhp ΓC (3.8a) Z Hq Hq hp Hq Hq Hq h hp b(µ − λ , u ) + γ(µn − λn )(−λn − σn(u )) ds ΓC Z + γ(µHq − λHq)(−λHq − σh(uhp)) ds g, µHq − λHq ∀ µHq ∈ MH (F). t t t t 6 n n ΓC ΓC (3.8b)

hI Here γ is deﬁned on each element I as the constant γ = γ0 2 , with γ0 > 0 independent pI h of p . Note that the additional stabilization term vanishes for the solution of the continuous Hq hp problem as λ → λ and Shpu → Su.

3.2.1 Existence and uniqueness of the solution

In this section we show the existence and uniqueness of a solution to the stabilized formulation. We follow ideas of V.Lleras [45] for the h-version of stabilized FEM.

Lemma 3.1. [9][Coercivity] For γ0 suﬃciently small, there exists a constant C > 0 independent of h, p, H and q, such that Z Z hp hp h hp 2 h hp 2 hp 2 hp hShpv , v iΣ − γ(σn(v )) ds − γ(σt (v )) ds Ckv k 1 , ∀v ∈ Vhp > H 2 (Σ) ΓC ΓC (3.9)

Lemma 3.2. For γ0 small enough, Problem (3.8) admits a unique solution.

hp Hq Hq Proof. Problem (3.8) is equivalent to ﬁnding a saddle-point (u , λn , λt ) ∈ Vhp × MHq(F) which satisﬁes hp Hq hp Hq hp Hq hp Hq H Lγ(u , ν ) ≤ Lγ(u , λ ) ≤ Lγ(v , λ ) ∀v ∈ Vhp, ∀ν ∈ M (F), (3.10)

44 3.2 The stabilized mixed hp-BEM formulation with 1 L (vhp, νHq) = hS vhp, vhpi − L(vhp) + b(νHq, vhp) γ 2 hp Σ Z Z 1 Hq h hp 2 1 Hq h hp 2 − γ(νn + σn(v )) ds − γ(νt + σt (v )) ds. (3.11) 2 ΓC 2 ΓC

Hq Taking ν = 0 in (3.11) and using Lemma3.1 with γ0 small enough , we obtain 1 L (vhp, 0) = hS vhp, vhpi − L(vhp) + b(0, vhp) γ 2 hp Σ Z Z 1 h hp 2 1 h hp 2 − γ(σn(v )) ds − γ(σt (v )) ds 2 ΓC 2 ΓC C hp 2 hp ≥ kv k 1 −kLkkv k 1 (3.12) 2 H 2 (Σ) H 2 (Σ) That yields hp lim Lγ(v , 0) = +∞ (3.13) hp kv k 1 →∞ H 2 (Σ) Choosing vhp = 0 in (3.11) we obtain Z Z Hq 1 Hq 2 1 Hq 2 Lγ(0, ν ) = − γ(νn ) ds − γ(νt ) ds. 2 ΓC 2 ΓC and Hq lim Lγ(0, ν ) = −∞ (3.14) kνHqk →∞ L2(ΓC ) hp Hq Then, due to (3.13) (3.14), Lγ is stricly convex in v and stricly concave in ν .

The existence of the solution to problem (3.8) follows from the fact that Vhp and H H M (F) are two nonempty closed convex sets, Lγ(·, ·) is continuous on Vhp × M (F), hp H hp Lγ(v , ·) (resp. Lγ(·, ν )) is stricly concave (resp. stricly convex) for any v ∈ Vhp (resp. for any νHq ∈ MH (F)) (see [35]).

Hq hp Hq hp Hq Hq Let (λ1 , u1 ) and (λ2 , u2 ) be two solution of (3.8). Then, choosing µ1 = λ2 and Hq Hq µ2 = λ1 in (3.8b), we get

Z Hq Hq hp Hq Hq Hq h hp b(λ2 − λ1 , u1 ) + γ(λ2,n − λ1,n)(−λ1,n − σn(u1 )) ds ΓC Z Hq Hq Hq h hp D Hq HqE + γ(λ2,t − λ1,t )(−λ1,t − σt (u1 )) ds 6 g, λ2,n − λ1,n Γ ΓC C (3.15)

Z Hq Hq hp Hq Hq Hq h hp b(λ1 − λ2 , u2 ) + γ(λ1,n − λ2,n)(−λ2,n − σn(u2 )) ds ΓC Z Hq Hq Hq h hp D Hq HqE + γ(λ1,t − λ2,t )(−λ2,t − σt (u2 )) ds 6 g, λ1,n − λ2,n Γ ΓC C (3.16)

45 3 Stabilized mixed hp-BEM in Linear Elasticity

Adding the last two inequalities, we obtain Z Hq Hq hp hp Hq Hq Hq Hq h hp hp b(λ1 − λ2 , u1 − u2 ) − γ(λ1,n − λ2,n)(λ1,n − λ2,n + σn(u1 − u2 )) ds ΓC Z Hq Hq Hq Hq h hp hp − γ(λ1,t − λ2,t )(λ1,t − λ2,t + σt (u1 − u2 )) ds ≥ 0 ΓC (3.17)

hp hp hp Futhermore, subtracting the two (3.8a) with v = u1 − u2 implies hp hp hp hp Hq Hq hp hp 0 = hShp(u1 − u2 ), u1 − u2 iΣ + b(λ1 − λ2 , u1 − u2 ) Z Hq Hq h hp hp h hp hp − γ(λ1,n − λ2,n + σn(u1 − u2 ))σn(u1 − u2 ) ds ΓC Z Hq Hq h hp hp h hp hp − γ(λ1,t − λ2,t + σt (u1 − u2 ))σt (u1 − u2 ) ds ΓC Z hp hp hp hp h hp hp h hp hp ≥ hShp(u1 − u2 ), u1 − u2 iΣ − γσn(u1 − u2 )σn(u1 − u2 ) ds ΓC Z Z h hp hp h hp hp Hq Hq 2 − γσt (u1 − u2 )σt (u1 − u2 ) ds + γ(λ1 − λ2 ) ds ΓC ΓC hp hp 1 Hq Hq 2 2 2 ≥ Cku1 − u2 k 1 +kγ (λ1 − λ2 )kL (Γ ) (3.18) H 2 (Σ) 2 C The conformity in the primal variable implies the uniqueness of the solution of problem (3.8).

hp Let u ∈ V and u ∈ Vhp. As in [15], we deﬁne the following vectors 1 ψ := V −1(K + )u 2 1 ψhp := i V −1i∗ (K + )uhp (3.19) hp hp hp 2 1 ψ∗ := V −1(K + )uhp. hp 2 ∗ hp Lemma 3.3. For ψhp, ψ deﬁned in (3.19) there holds

∗ hp hV (ψhp − ψ ), φi = 0 ∀φ ∈ Whp (3.20) Lemma 3.4. (Galerkin orthogonality) Let u ∈ V be the solution of the continuous hp problem (3.4) and u ∈ Vhp the solution of the discrete problem. There holds Z hp hp Hq hp Hq h hp h hp hSu − Shpu , v iΣ + b(λ − λ , v ) − γ(−λn − σn(u ))σn(v ) ds ΓC Z Hq h hp h hp hp − γ(−λt − σt (u ))σt (v ) ds = 0 ∀v ∈ Vhp. ΓC (3.21)

Proof. We choose v ∈ Vhp ⊂ V in (3.4a) and subtract (3.4a) from the discrete formulation (3.8a).

46 3.3 A priori error analysis for frictional contact problem

3.3 A priori error analysis for frictional contact problem

Lemma 3.5. Let u ∈ V, λ ∈ M solve the saddle point problem (3.4) and, let uhp ∈ Hq Vhp and λ ∈ Mhp(F) be the solution of the discrete problem (3.8), we assume that λ ∈ L2(ΓC ). Then there holds Z 1 Hq 2 1 Hq 2 Hq kγ 2 (λ − λ )k + kγ 2 (λ − λ )k ≤ (λ − µ )u ds n n L2(ΓC ) t t L2(ΓC ) n n n ΓC Z Hq hp Hq hp + (λn − µn )(un + γ(−λn − σn(u )) ds ΓC Z Hq hp Hq hp − γ(λn − λn )σn(u − u ) ds − hλn − λn , un − uni ΓC Z Z Hq Hq hp Hq hp + (λt − µt)ut ds + (λt − µt )(ut + γ(−λt − σt(u )) ds ΓC ΓC Z Hq hp Hq hp − γ(λt − λt )σt(u − u ) ds − hλt − λt , ut − uti ΓC Z Z Hq Hq n hp Hq Hq t hp − γ(µn − λn )Eh (u ) ds − γ(µt − λt )Eh(u ) ds ΓC ΓC (3.22) where n hp hp t hp hp Eh (u ) = (S − Shp)u · n|ΓC ,Eh(u ) = (S − Shp)u · t|ΓC (3.23)

Proof. Recall that λn = −σn(u) and λt = −σt(u).

The inequality in (3.8b) is equivalent to the following conditions: Z Hq Hq hp Hq Hq Hq h hp hµn − λn , un i + γ(µn − λn )(−λn − σn(u )) ds ≤ 0 (3.24) ΓC Z Hq Hq hp Hq Hq Hq h hp hµt − λt , ut i + γ(µt − λt )(−λt − σt (u )) ds ≤ 0. (3.25) ΓC Note that Z Z Z 1 Hq 2 2 Hq Hq 2 kγ 2 (λ − λ )k = γλ ds − 2 γλ λ ds + γ(λ ) ds (3.26) n n L2(ΓC ) n n n n ΓC ΓC ΓC Z Z Z 1 Hq 2 2 Hq Hq 2 kγ 2 (λ − λ )k = γλ ds − 2 γλ λ ds + γ(λ ) ds (3.27) t t L2(ΓC ) t t t t ΓC ΓC ΓC Using (3.24) and (3.25) we get Z Z Z Hq 2 Hq Hq Hq Hq hp γ(λn ) ds ≤ γλn µn ds − (µn − λn )un ds ΓC ΓC ΓC Z Hq Hq h hp + γ(µn − λn )σn(u ) ds ΓC Z Z Z Hq 2 Hq Hq Hq Hq hp γ(λt ) ds ≤ γλt µt ds − (µt − λt )ut ds ΓC ΓC ΓC Z Hq Hq h hp + γ(µt − λt )σt (u ) ds. ΓC

47 3 Stabilized mixed hp-BEM in Linear Elasticity

From the second equation of the continuous problem (3.4b), we have for all (λn, λt) ∈ Mn × Mt(F): Z Z Z Z 2 γ(λn) ds ≤ γλnµn ds − (µn − λn)un ds + γ(µn − λn)σn(u) ds ΓC ΓC ΓC ΓC Z Z Z Z 2 γ(λt) ds ≤ γλtµt ds − (µt − λt)ut ds + γ(µt − λt)σt(u) ds. ΓC ΓC ΓC ΓC

This gives

1 Hq 2 1 Hq 2 kγ 2 (λ − λ )k +kγ 2 (λ − λ )k ≤ n n L2(ΓC ) t t L2(ΓC ) Z Z Z Hq Hq Hq γ(µn − λn )λn ds + γ(µn − λn)λn ds + (λn − µn)un ds ΓC ΓC ΓC Z Z Z Hq Hq hp Hq Hq h hp + γ(µn − λn)σn(u) ds + (λn − µn )un ds + γ(µn − λn )σn(u ) ds ΓC ΓC ΓC Z Z Z Hq Hq Hq + γ(µt − λt )λt ds + γ(µt − λt)λt ds + (λt − µt)ut ds ΓC ΓC ΓC Z Z Z Hq Hq hp Hq Hq h hp + γ(µt − λt)σt(u) ds + (λt − µt )ut ds + γ(µt − λt )σt (u ) ds ΓC ΓC ΓC

Z Z Z Hq Hq Hq = γ(µn − λn )λn ds + γ(µn − λn)λn ds + (λn − µn)un ds ΓC ΓC ΓC Z Z Hq Hq hp + γ(µn − λn)σn(u) ds + (λn − µn )un ds ΓC ΓC Z Z Hq Hq hp Hq Hq n hp + γ(µn − λn )σn(u ) ds − γ(µn − λn )Eh (u ) ds ΓC ΓC Z Z Z Hq Hq Hq + γ(µt − λt )λt ds + γ(µt − λt)λt ds + (λt − µt)ut ds ΓC ΓC ΓC Z Z Hq Hq hp + γ(µt − λt)σt(u) ds + (λt − µt )ut ds ΓC ΓC Z Z Hq Hq hp Hq Hq t hp + γ(µt − λt )σt(u ) ds − γ(µt − λt )Eh(u ) ds ΓC ΓC

Z Z Hq Hq hp Hq hp = (λn − µn)un ds + (λn − µn )(un + γ(−λn − σn(u )) ds ΓC ΓC Z Z Hq hp Hq hp Hq − γ(λn − λn )σn(u − u ) ds − hλn − λn , un − uni + (λt − µt)ut ds ΓC ΓC Z Z Hq hp Hq hp Hq hp + (λt − µt )(ut + γ(−λt − σt(u )) ds − γ(λt − λt )σt(u − u ) ds ΓC ΓC Z Z Hq hp Hq Hq n hp Hq Hq t hp − hλt − λt , ut − uti − γ(µn − λn )Eh (u ) ds − γ(µt − λt )Eh(u ) ds. ΓC ΓC

48 3.3 A priori error analysis for frictional contact problem

Lemma 3.6. There exists a constant C > 0, independent of h, p, such that

hp ku k 1 ≤ C (3.28) H 2 (Σ)

In particular, for u 6= 0, there exists a constant Cu such that hp ku k 1 ≤ Cukuk 1 (3.29) H 2 (Σ) H 2 (Σ)

Proof. Choosing vhp = uhp in (3.8a), we obtain Z Z hp hp Hq hp Hq h hp h hp hShpu , u iΣ + λn un ds + γ(−λn − σn(u ))σn(u ) ds ΓC ΓC Z Z Hq hp Hq h hp h hp hp + λt ut ds + γ(−λt − σt (u ))σt (u ) ds = L(u ) (3.30) ΓC ΓC Hq Hq Hq Choosing µn = 0 and µt = λt in (3.8b), we get Z Z Hq hp Hq Hq h hp − λn un ds − λn (−λn − σn(u )) ≤ 0 (3.31) ΓC ΓC Hq Hq Hq Hq Now we choose µn = 2λn and µt = λt in (3.8b), we obtain Z Z Hq hp Hq Hq h hp λn un ds + λn (−λn − σn(u )) ≤ 0 (3.32) ΓC ΓC Combining (3.31) and (3.32), we obtain Z Z Hq hp Hq Hq h hp λn un ds + γλn (−λn − σn(u )) = 0 (3.33) ΓC ΓC Similarly, we get Z Z Hq hp Hq Hq h hp λt ut ds + γλt (−λt − σt (u )) = 0 (3.34) ΓC ΓC Using (3.30), (3.33), and (3.34), we have Z Z hp hp hp Hq hp Hq h hp h hp L(u ) = hShpu , u iΣ + λn un ds + γ(−λn − σn(u ))σn(u ) ds ΓC ΓC Z Z Hq hp Hq h hp h hp + λt ut ds + γ(−λt − σt (u ))σt (u ) ds ΓC ΓC Z Z hp hp Hq 2 h hp 2 = hShpu , u iΣ + γ(λn ) ds − γ(σn(u )) ds ΓC ΓC Z Z Hq 2 h hp 2 + γ(λt ) ds − γ(σt (u )) ds (3.35) ΓC ΓC From Lemma 3.1 and (3.35), we obtain Z Z hp 2 h hp 2 h hp 2 hp Cku k 1 ≤ hShpu, uiΓ − γ(σn(u )) ds − γ(σt (u )) ds ≤ L(u ) H 2 (Σ) ΓC ΓC hp ≤ ku k 1 H 2 (Σ) (3.36)

49 3 Stabilized mixed hp-BEM in Linear Elasticity

Theorem 3.1. Let (u, λ) ∈ V × M be a solution of the problem (3.4) such that u ∈ 1 hp Hq H˜ (Σ) and λ ∈ L2(ΓC ). Let t ∈ L2(Σ) . Let (u , λ ) be the solution of the discrete problem (3.8). Then there exists a constant C > 0 independent of h, H, p and q, such that

1 1 Hq hp 2 hp 2 2 Hq 2 2 2 ku − u k 1 +kψ − ψ k − 1 +kγ (λn − λn )kL (Γ )+kγ (λt − λt )kL (Γ ) H˜ 2 (Σ) H˜ 2 (Γ) 2 C 2 C " hp 2 1 hp 2 ≤ C inf ku − v k 1 +kγ 2 σn(u − v )k hp 2 L2(ΓC ) v ∈Vhp H˜ (Σ) Hq hp hp +b(λ , u − v ) + ku − v kL2(Σ) Z Z hp hp hp hp hp hp + γ(λn + σn(u ))σn(u − v ) ds + γ(λt + σt(u ))σt(u − v ) ds ΓC ΓC Z Z Hq hp h hp hp Hq hp h hp hp − γ(λn + σn(u ))En(u − v ) ds − γ(λt + σt(u ))Et (u − v ) ds ΓC ΓC Z Z h hp hp hp h hp hp hp + γEn(u )σn(u − v ) ds + γEt (u )σt(u − v ) ds ΓC ΓC Z Z h hp h hp hp h hp h hp hp + γEn(u )En(u − v ) ds + γEt (u )Et (u − v ) ds ΓC ΓC hp hp + inf b(λ, v − u ) + ku − vkL (Σ) v∈V 2 Z 2 Hq + inf kψ − φk 1 + inf (λn − µn)un ds φ∈W ˜ − 2 µ ∈M hp H (Γ) n n ΓC Z Hq hp Hq hp + inf (µ − λn)(u + γ(−λ − σn(u )) ds Hq n n n µn ∈Mn,Hq ΓC Z Z Hq Hq hp Hq hp + inf (λt − µt)ut ds + inf (µt − λt)(ut + γ(−λt − σt(u )) ds µt∈Mt(F) Hq ΓC µt ∈Mt,Hq(F) ΓC Z Z # + inf γ(λHq − µHq)En(uhp) ds + inf γ(λHq − µHq)Et (uhp) ds Hq n n h Hq t t h µn ∈Mn,Hq ΓC µt ∈Mt,Hq(F) ΓC

Proof. From Lemma 2.10 we obtain

hp 2 2 hp hp ∗ hp hp ku − u kW +kψ − ψhpkV = hSu − Shpu , u − u i + hV (ψhp − ψ ), ψ − ψ i.

Since we have with Ehp = S − Shp

hp hp hp hp hp Su − Shpu = S(u − u ) + Ehpu = S(u − u ) + Ehp(u − u) + Ehpu, (3.37) we obtain

hp 2 2 hp hp ∗ hp hp ku − u kW +kψ − ψhpkV = hSu − Shpu , u − u i + hV (ψhp − ψ ), ψ − ψ i = hS(u − uhp), u − vhpi + hSuhp, uhp − vhpi + hSu, vhp − uhpi hp hp ∗ hp hp + hEhpu , u − u i + hV (ψhp − ψ ), ψ − ψ i. (3.38)

50 3.3 A priori error analysis for frictional contact problem

Using (3.8a) and the deﬁnition of Shp, we get

hp hp hp hp hp hp Hq hp hp hp hp hSu , u − v i = hEhpu , u − v i − b(λ , u − v ) + ht, u − v i Z Z Hq h hp h hp hp Hq h hp h hp hp + γ(λn + σn(u ))σn(u − v ) ds + γ(λt + σt (u ))σt (u − v ) ds ΓC ΓC (3.39)

Using (3.4), (3.38) and (3.39), we obtain

hp 2 hp 2 hp hp hp hp CW ku − u k 1 +CV kψ − ψ k − 1 ≤ hS(u − u ), u − v i + hSu, v − u i H˜ 2 (Σ) H˜ 2 (Γ)

hp hp hp hp hp Hq hp hp hp hp + hEhpu , u − u i + hEhpu , u − v i − b(λ , u − v ) + ht, u − v i

∗ hp hp − hSu, u − vi − b(λ, u − v) + ht, u − vi + hV (ψhp − ψ ), ψ − ψ i

Z Z Hq hp hp hp Hq hp hp hp + γ(λn + σn(u ))σn(u − v ) ds + γ(λt + σt(u ))σt(u − v ) ds ΓC ΓC

hp hp hp hp hp hp ≤ hS(u − u ), u − v i + hEhpu , u − v i + ht − Su, u − vi + ht − Su, u − v i

Hq hp hp ∗ hp hp + b(λ, v − u) − b(λ , u − v ) + hV (ψhp − ψ ), ψ − ψ i

Z Z Hq h hp h hp hp Hq h hp h hp hp + γ(λn + σn(u ))σn(u − v ) ds + γ(λt + σt (u ))σt (u − v ) ds ΓC ΓC (3.40)

We have b(λ, v − u) − b(λHq, uhp − vhp) − b(λ − λHq, uhp − u) = b(λ, v − uhp) − b(λHq, u − vhp) (3.41)

From (3.40) and (3.41), we obtain

hp 2 hp 2 Hq hp CW ku − u k 1 +CV kψ − ψ k − 1 −hλ − λ , u − ui ≤ H˜ 2 (Σ) H˜ 2 (Γ)

hp hp hp hp hp hp hS(u − u ), u − v i + hEhpu , u − v i + ht − Su, u − vi + ht − Su, u − v i

hp Hq hp ∗ hp hp + b(λ, v − u ) − b(λ , u − v ) + hV (ψhp − ψ ), ψ − ψ i

Z Z Hq h hp h hp hp Hq h hp h hp hp + γ(λn + σn(u ))σn(u − v ) ds + γ(λt + σt (u ))σt (u − v ) ds ΓC ΓC (3.42)

51 3 Stabilized mixed hp-BEM in Linear Elasticity

Using Lemma 3.5 and (3.42) we have

1 1 Hq hp 2 hp 2 2 Hq 2 2 2 CW ku − u k 1 +CV kψ − ψ k − 1 +kγ (λn − λn )kL (Γ )+kγ (λt − λt )kL (Γ ) H˜ 2 (Σ) H˜ 2 (Γ) 2 C 2 C hp hp hp hp hp hp ≤ hS(u − u ), u − v i + hEhpu , u − v i + ht − Su, u − vi + ht − Su, u − v i hp Hq hp ∗ hp hp + b(λ, v − u ) − b(λ , u − v ) + hV (ψhp − ψ ), ψ − ψ i Z Z Hq h hp h hp hp Hq h hp h hp hp + γ(λn + σn(u ))σn(u − v ) ds + γ(λt + σt (u ))σt (u − v ) ds ΓC ΓC Z Z Hq Hq hp Hq hp + (λn − µn)un ds + (λn − µn )(un + γ(−λn − σn(u )) ds ΓC ΓC Z Z Hq hp Hq hp − γ(λn − λn )σn(u − u ) ds − γ(λt − λt )σt(u − u ) ds ΓC ΓC Z Z Hq Hq hp Hq hp + (λt − µt)ut ds + (λt − µt )(ut + γ(−λt − σt(u )) ds ΓC ΓC Z Z Hq Hq n hp Hq Hq t hp + γ(λn − µn )Eh (u ) ds + γ(λt − µt )Eh(u ) ds (3.43) ΓC ΓC

Therefore

1 1 Hq hp 2 hp 2 2 Hq 2 2 2 CW ku − u k 1 +CV kψ − ψ k − 1 +kγ (λn − λn )kL (Γ )+kγ (λt − λt )kL (Γ ) H˜ 2 (Σ) H˜ 2 (Γ) 2 C 2 C hp hp hp hp hp hp ≤ hS(u − u ), u − v i + hEhpu , u − v i + ht − Su, u − vi + ht − Su, u − v i hp Hq hp ∗ hp hp + b(λ, v − u ) − b(λ , u − v ) + hV (ψhp − ψ ), ψ − ψ i Z Z hp hp hp hp hp hp γ(λn + σn(u ))σn(u − v ) ds + γ(λt + σt(u ))σt(u − v ) ds ΓC ΓC Z Z Hq Hq hp Hq hp + (λn − µn)un ds + (λn − µn )(un + γ(−λn − σn(u )) ds ΓC ΓC Z Z Hq hp Hq hp − γ(λn − λn )σn(u − v ) ds − γ(λt − λt )σt(u − v ) ds ΓC ΓC Z Z Hq Hq hp Hq hp + (λt − µt)ut ds + (λt − µt )(ut + γ(−λt − σt(u )) ds ΓC ΓC Z Z Hq hp h hp hp Hq hp h hp hp − γ(λn + σn(u ))En(u − v ) ds − γ(λt + σt(u ))Et (u − v ) ds ΓC ΓC Z Z h hp hp hp h hp hp hp − γEn(u )σn(u − v ) ds − γEt (u )σt(u − v ) ds ΓC ΓC Z Z h hp h hp hp h hp h hp hp + γEn(u )En(u − v ) ds + γEt (u )Et (u − v ) ds ΓC ΓC Z Z Hq Hq h hp Hq Hq t hp + γ(λn − µn )En(u ) ds + γ(λt − µt )Eh(u ) ds (3.44) ΓC ΓC

Using (3.37), Cauchy Schwarz inequality, and the continuity of Shp and Ehp, there holds for all φ ∈ Whp

hp hp hp hp A = hS(u − u ), u − v i ≤ CSku − u k 1 ku − v k 1 (3.45) H˜ 2 (Σ) H˜ 2 (Σ)

52 3.3 A priori error analysis for frictional contact problem

Using Lemma2.6and Cauchy Schwarz inequality, we have

hp hp hp hp B = hEhpu , u − v i = hEhp(u + u − u), u − v i ≤ hp hp hp kEhpuk − 1 ku − v k 1 +CE ku − u k 1 ku − v k 1 H 2 (Σ) H˜ 2 (Σ) hp H˜ 2 (Σ) H˜ 2 (Σ) hp hp hp hp ≤ C0kψ − φ k − 1 ku − v k 1 +CE ku − u k 1 ku − v k 1 H˜ 2 (Γ) H˜ 2 (Σ) hp H˜ 2 (Σ) H˜ 2 (Σ) (3.46)

hp for φ ∈ Whp.

hp From Lemma3.3 and (3.19) follows with φ ∈ Whp that

∗ hp hp C = hV (ψh − ψ ), ψ − ψ i ∗ hp hp hp = hV (ψh − ψ), ψ − φ i + hV (ψ − ψ ), ψ − φ i 1 h hp ≤ (CK + )ku − u k 1 kψ − φ k − 1 2 H˜ 2 (Σ) H˜ 2 (Γ) hp hp + CV kψ − ψ k − 1 kψ − φ k − 1 (3.47) H˜ 2 (Γ) H˜ 2 (Γ)

Employing Young’s inequality, we obtain

1 2 hp 2 hp 2 A ≤ CSku − v k 1 + ku − u k 1 (3.48) 2 H˜ 2 (Σ) 2 H˜ 2 (Σ)

1 2 hp 2 hp 2 B ≤ CE ku − v k 1 + ku − u k 1 2 hp H˜ 2 (Σ) 2 H˜ 2 (Σ)

C0 hp 2 C0 hp 2 + kψ − φ k − 1 + ku − v k 1 (3.49) 2 H˜ 2 (Γ) 2 H˜ 2 (Σ)

1 1 2 2 hp 2 C ≤ (CK + ) kψ − φk − 1 + ku − u k 1 2 2 H˜ 2 (Γ) 2 H˜ 2 (Σ) 1 2 hp 2 hp 2 + CV kψ − φ k − 1 + kψ − ψ k − 1 (3.50) 2 H˜ 2 (Γ) 2 H˜ 2 (Γ)

Since t − Su ∈ L2(Σ), we obtain

D = ht − Su, uhp − vi + ht − Su, u − vhpi hp hp ≤ kt − SukL2(Σ)(ku − vkL2(Σ)+ku − v kL2(Σ)) (3.51)

Employing Cauchy Schwarz and Young’s inequality, we obtain

Z 1 1 Hq hp 2 Hq 2 hp En = γ(λn − λn )σn(u − v ) ds ≤ kγ (λn − λn )kL2(ΓC )kγ σn(u − v )kL2(ΓC ) ΓC 1 Hq 2 1 1 hp 2 ≤ kγ 2 (λn − λ )k + kγ 2 σn(u − v )k (3.52) 2 n L2(ΓC ) 2 L2(ΓC )

53 3 Stabilized mixed hp-BEM in Linear Elasticity

Z Hq 1 Hq 1 hp 2 2 hp Et = γ(λt − λt )σt(u − v ) ds ≤ kγ (λt − λt )kL2(ΓC )kγ σt(u − v )kL2(ΓC ) ΓC 1 Hq 2 1 1 hp 2 ≤ kγ 2 (λt − λ )k + kγ 2 σt(u − v )k (3.53) 2 t L2(ΓC ) 2 L2(ΓC )

Using (3.44), and (3.48)-(3.53), we get

hp 2 hp 2 α1ku − u k 1 +α2kψ − ψ k − 1 H˜ 2 (Σ) H˜ 2 (Γ)

1 Hq 2 1 Hq 2 + α kγ 2 (λ − λ )k +α kγ 2 (λ − λ )k 3 n n L2(ΓC ) 4 t t L2(ΓC )

hp 2 hp 2 ≤ α5ku − v k 1 +α6kψ − φ k − 1 H˜ 2 (Σ) H˜ 2 (Γ)

1 1 hp 2 1 1 hp 2 + kγ 2 σn(u − v )k + kγ 2 σt(u − v )k L2(ΓC ) L2(ΓC )

hp hp hp Hq hp + kt − SukL2(Σ)(ku − vkL2(Σ)+ku − v kL2(Σ)) + b(λ, v − u ) − b(λ , u − v )

Z Z Hq Hq hp Hq hp + (λn − µn)un ds − (λn − µn )(un + γ(−λn − σn(u )) ds ΓC ΓC

Z Z Hq Hq hp Hq hp + (λt − µt)ut ds − (λt − µt )(ut + γ(−λt − σt(u )) ds ΓC ΓC

Z Z hp hp hp hp hp hp + γ(λn + σn(u ))σn(u − v ) ds + γ(λt + σt(u ))σt(u − v ) ds ΓC ΓC

Z Z Hq hp n hp hp Hq hp t hp hp − γ(λn + σn(u ))Ehp(u − v ) ds − γ(λt + σt(u ))Ehp(u − v ) ds ΓC ΓC

Z Z n hp hp hp t hp hp hp + γEhp(u )σn(u − v ) ds + γEhp(u )σt(u − v ) ds ΓC ΓC

Z Z n hp n hp hp t hp t hp hp + γEhp(u )Ehp(u − v ) ds + γEhp(u )Ehp(u − v ) ds ΓC ΓC

Z Z Hq Hq n hp Hq Hq t hp + γ(λn − µn )Ehp(u ) ds + γ(λt − µt )Ehp(u ) ds, ΓC ΓC

54 3.3 A priori error analysis for frictional contact problem where the constants

α1 = 2CW − 3

α2 = 2CV −

α3 = 2 −

α4 = 2 − 2 2 C CE α = S + hp + C 5 0 1 1 1 α = C + (C + )2 + C2 6 0 K 2 V are independent of h, H, p and q, α1, α2, α3 and α4 are positive if is small enough. The estimate of the theorem follows immediately.

Theorem 3.2. Let (u, λ) ∈ V × M be the solution of the problem (3.4) and (uhp, λHq) the solution of the discrete problem (3.8) with t ∈ L2(ΓN ) and g = 0. Assume that ˜ 1+ν ν 1 u ∈ H (Σ), λ ∈ H (ΓC ) for some ν ∈ [0, 2 ]. Suppose that

kλ k ν +kλ k ν +kFk ≤ kuk 1+ν . n H (ΓC ) t H (ΓC ) L2(ΓC ) H˜ (Σ) Then there exists a constant C > 0 independent of h and p, such that 1 1 Hq hp hp 2 Hq 2 ku − u k 1 +kψ − ψ k − 1 +kγ (λn − λn )kL (Γ )+kγ (λt − λt )kL (Γ ) H˜ 2 (Σ) H˜ 2 (Γ) 2 C 2 C ν+ 1 1 ν ! H 2 h 2 H ≤ C + kuk 1+ν ν+ 1 ν H˜ (Σ) q 2 pq Z Z Hq Hq + inf (λn − µn)un ds + inf (λt − µt)ut ds (3.54) µ ∈M µ ∈M (F) n n ΓC t t ΓC

Proof. The a priori estimate follows from the estimate in Theorem (3.1) with v = uhp.

We estimate the term

Hq hp Hq hp Hq hp b(λ , u − v ) = hλn , un − vn i + hλt , ut − vt i (3.55) Employing Cauchy Schwarz and Young’s inequality, we have Z Z Hq hp Hq hp hp hλn , un − vn i = (λn − λn)(un − vn ) ds + λn(un − vn ) ds ΓC ΓC Hq hp hp ≤ kλn − λnkL2(ΓC )kun − vn kL2(ΓC )+kλnkL2(ΓC )kun − vn kL2(ΓC ) 2 1 Hq 2 1 p hp 2 hp ≤ kγ 2 (λ − λ )k + ku − v k +kλ k ku − v k n n L2(Γ ) n n L2(Γ ) n L2(ΓC ) n n L2(ΓC ) 2 C 2γ0 h C 2ν+1 ν+1 1 Hq 2 1 h 2 h 2 ≤ kγ 2 (λ − λn)k + kuk 1+ν + kuk 1+ν (3.56) 2 n L2(ΓC ) 2 p2ν H (Γ) pν+1 H (Γ) Similarly to (2.101), we get 2ν+1 ν+1 Hq hp 1 Hq 2 1 h 2 h 2 hλ , ut − v i ≤ ≤ kγ 2 (λ − λt)k + kuk 1+ν + kuk 1+ν t t 2 t L2(ΓC ) 2 p2ν H (Γ) pν+1 H (Γ) (3.57)

55 3 Stabilized mixed hp-BEM in Linear Elasticity

Combining (2.101) and (3.57), we obtain

Hq hp 1 Hq 2 1 Hq 2 b(λ , u − v ) ≤ kγ 2 (λ − λn)k + kγ 2 (λ − λt)k 2 n L2(ΓC ) 2 t L2(ΓC ) 1 h2ν+1 hν+1 + kuk2 + kuk2 (3.58) 2 p2ν H1+ν (Γ) pν+1 H1+ν (Γ)

Now we estimate the terms

1 hp 2 2 hp 2 A2 = ku − v k 1 and A3 = kγ σn(u − v )kL (Γ ) H˜ 2 (Σ) 2 C

hp We choose v = Ihpu and use the aproximation property of the Lagrange interpolation operator:

1+2ν hp 2 2 h 2 A2 = ku − v k 1 = ku − Ihpuk 1 ≤ C 1+2ν kuk ˜ 1+ν (3.59) H˜ 2 (Σ) H˜ 2 (Σ) p H (Σ) we have

2ν hp 2 2 h 2 ku − v k 1 = ku − Ihpuk 1 ≤ C kuk 1+ν (3.60) H (Γ) H (Γ) p H˜ (Σ)

Employing the continuity condition of the Dirichlet-to-Neumann operator and (3.60) we obtain

1 hp 2 1 hp 2 A = kγ 2 σ (u − v )k ≤ αkγ 2 (u − v )k 1 3 n L2(ΓC ) H (Γ) h = αγ ku − I uk2 0 p2 hp H1(Γ) 1+2ν h 2 = Cαγ0 kuk 1+ν . (3.61) p2+2ν H˜ (Σ)

We now estimate the term

Z hp hp hp A4,n = γ(σn(u ) + λn)σn(u − v ) ds. ΓC

hp hp Recall that λn = −σn(u). Since u −v ∈ Vhp, we can apply the hp-inverse inequality

h hp hp 2 h hp hp 2 kσn(u − v )k ≤ α ku − v k 1 p2 L2(ΓC ) p2 H (Γ) hp hp 2 ≤ αku − v k 1 (3.62) H˜ 2 (Σ)

56 3.3 A priori error analysis for frictional contact problem

Using Cauchy Schwarz, Young’s inequality, (3.59),(3.61) and (3.62), we obtain

1 h hp hp 2 1 h hp 2 A4,n ≤ γ0 kσn(u − v )k + γ0 kσn(u − u )k 2 p2 L2(ΓC ) 2 p2 L2(ΓC )

h hp hp 2 1 h hp 2 ≤ γ0 kσn(u − v )k + γ0 kσn(u − v )k p2 L2(ΓC ) 2 p2 L2(ΓC )

1+2ν h 2 hp 2 hp 2 ≤ Cαγ0 2+2ν kuk ˜ 1+ν +2αγ0ku − v k 1 +2αγ0ku − u k 1 p H (Σ) H˜ 2 (Σ) H˜ 2 (Σ)

1+2ν 1+2ν ˇ h 2 h 2 hp 2 ≤ C 2+2ν kuk ˜ 1+ν + 1+2ν kuk ˜ 1+ν +γ0ku − u k 1 . (3.63) p H (Σ) p H (Σ) H˜ 2 (Σ) Similar to (3.63), we get Z hp hp hp A4,t = γ(σt(u ) + λt)σt(u − w ) ds ΓC 1+2ν 1+2ν ! ˇ h 2 h 2 hp 2 ≤ C 2+2ν kuk ˜ 1+ν + kuk ˜ 1+ν +γ0ku − u k 1 (3.64) p H (Σ) p H (Σ) H˜ 2 (Σ) We now estimate the term Z Hq hp Hq hp A5,n = inf (µ − λn)(u + γ(−λ − σn(u )) ds (3.65) Hq n n n µn ∈Mn,Hq ΓC

As in Chapter2, let πMhp be the L2-projection operator mapping MHq deﬁned by ( πMn,Hq : L2(ΓC ) −→ Mn,Hq πMHq = (3.66) πMt,Hq : L2(ΓC ) −→ Mt,Hq.

Hq Choosing µn = πMn,Hq λn, we obtain Z Z hp Hq hp A5,n ≤ (πMn,Hq λn − λn)un ds + γ(πMn,Hq λn − λn)(−λn − σn(u )) ds. ΓC ΓC (3.67) The estimate of the ﬁrst term in (3.67) gives, using (2.112): Z hp A5,1,n = (πMn,Hq λn − λn)un ds ΓC Z Z hp = (πMn,Hq λn − λn)(un − un) ds + (πMn,Hq λn − λn)(un − πMn,Hq un) ds ΓC ΓC hp ≤ kπMn,Hq λn − λnk − 1 kun − unk 1 H 2 (ΓC ) H 2 (ΓC )

+ kπMn,Hq λn − λnkL2(ΓC )kun − πMn,Hq unkL2(ΓC ) 1 +ν 1+2ν ! H 2 hp H ≤ C kλnkHν (Γ )ku − u k 1 + kλnkHν (Γ )kuk 1+ν 1 +ν C ˜ 2 1+2ν C H˜ (Σ) q 2 H (Σ) q (3.68)

57 3 Stabilized mixed hp-BEM in Linear Elasticity

We employ Young’s inequality to obtain

2ν+1 hp 2 H 2 A5,1,n ≤ C ku − u k 1 + 2ν+1 kuk ˜ 1+ν (3.69) H˜ 2 (Σ) q H (Σ)

Now we estimate the second integral term Z Hq Hq hp A5,2,n = inf γ(µ − λn)(−λ − σn(u )) ds. (3.70) Hq n n µn ∈Mn,Hq ΓC

We have Z Hq A5,2,n ≤ γ(πMn,Hq λn − λn)(−λn + λn) ds ΓC Z

+ γ(πMn,Hq λn − λn)(σn(u − Ihpu)) ds ΓC Z hp + γ(πMn,Hq λn − λn)(σn(Ihpu − u )) ds ΓC 1 ν 1 2 1 2 h H Hq ≤ Cγ kλ k ν kγ 2 (λ − λ )k 0 p2 qν n H (ΓC ) n n L2(ΓC ) 1 ν 1 2 1 2 h H + Cγ kλ k ν kγ 2 σ (u − I u)k 0 p2 qν n H (ΓC ) n hp L2(ΓC ) 1 ν 1 2 1 2 h H hp + Cγ kλ k ν kγ 2 σ (I u − u )k . (3.71) 0 p2 qν n H (ΓC ) n hp L2(ΓC )

Using Young’s inequality, (3.61), (3.60) and (3.63), we obtain

2ν hH 2 1 Hq 2 A5,2,n ≤ C kuk 1+ν +kγ 2 (λn − λ )k p2q2ν H˜ (Σ) n L2(ΓC ) 1+2ν h 2 hp 2 + 1+2ν kuk ˜ 1+ν +αγ0ku − u k 1 . (3.72) p H (Σ) H˜ 2 (Σ)

Finally we have

1 2 Hq 2 hp 2 A5,n ≤ C kγ (λn − λn )kL (Γ )+γ0ku − u k 1 2 C H˜ 2 (Σ) 2ν+1 2ν 2ν+1 H 2 hH 2 h 2 + kuk 1+ν + kuk 1+ν + kuk 1+ν . (3.73) q2ν+1 H˜ (Σ) p2q2ν H˜ (Σ) p2ν+1 H˜ (Σ)

Similarly to (3.73), we obtain Z Hq hp Hq hp A5,t = inf (µ − λt)(u + γ(−λ − σt(u )) ds Hq t t t µt ∈Mt,Hq(F) ΓC 1 Hq 2 2 hp 2 ≤ C kγ (λt − λt )kL (Γ )+γ0ku − u k 1 2 C H˜ 2 (Σ) 2ν+1 2ν 2ν+1 H 2 hH 2 h 2 + kuk 1+ν + kuk 1+ν + kuk 1+ν . (3.74) q2ν+1 H˜ (Σ) p2q2ν H˜ (Σ) p2ν+1 H˜ (Σ)

58 3.3 A priori error analysis for frictional contact problem

We now estimate the term Z Hq hp h hp hp A6,n = inf γ(−λ − σn(u ))E (u − v ) ds, (3.75) hp n n v ∈Vhp ΓC which we write as Z Z Hq h hp hp hp h hp hp A6,n = inf γ(λn − λ )E (u − v ) ds + γσn(u − u ))E (u − v ) ds hp n n n v ∈Vhp ΓC ΓC (3.76)

From a computation as in(3.63), we have

1+2ν 1+2ν h hp 2 h 2 h 2 hp 2 γ kσ (u − u )k ≤ C kuk 1+ν + kuk 1+ν +γ ku − u k 1 0 2 n L2(ΓC ) 2+2ν ˜ 1+2ν ˜ 0 p p H (Σ) p H (Σ) H˜ 2 (Σ) (3.77) and with continuity of Ehp = S − Shp

1+2ν h h hp hp 2 h 2 hp 2 γ kE (u − v )k ≤ C kuk 1+ν +γ ku − u k 1 . (3.78) 0 2 n L2(ΓC ) 1+2ν ˜ 0 p p H (Σ) H˜ 2 (Σ)

We obtain

1+2ν h 2 hp 2 1 Hq 2 A ≤ C kuk 1+ν +γ ku − u k 1 +kγ 2 (λ − λ )k (3.79) 6,n 1+2ν ˜ 0 n n L2(ΓC ) p H (Σ) H˜ 2 (Σ)

Similarly to (3.79), we get Z Hq hp h hp hp A6,t = inf γ(−λ − σt(u ))E (u − v ) ds hp t t v ∈Vhp ΓC 1+2ν h 2 hp 2 1 Hq 2 ≤ C kuk 1+ν +γ ku − u k 1 +kγ 2 (λ − λ )k (3.80) 1+2ν ˜ 0 t t L2(ΓC ) p H (Σ) H˜ 2 (Σ)

Consider now Z h hp hp hp A7,n = inf γE (u )σn(u − v ) ds (3.81) hp n v ∈Vhp ΓC

We have Z Z h hp hp hp h hp hp A7,n ≤ γEn(u − u)σn(u − v ) ds + γEn(u)σn(u − v ) ds ΓC ΓC (3.82)

Using Cauchy Schwarz, Young’s inequality, we have

h hp 2 1 hp hp 2 A7,n ≤ kEn(u − u)k − 1 + kγσn(u − v )k 1 2 H 2 (ΓC ) 2 H 2 (ΓC ) 1 h 2 1 hp hp 2 + kEn(u)k − 1 + kγσn(u − v )k 1 (3.83) 2 H 2 (ΓC ) 2 H 2 (ΓC )

59 3 Stabilized mixed hp-BEM in Linear Elasticity

Employing the continuity condition of the Dirichlet-to-Neumann operator and the inverse inequality, we get

hp hp 2 2 hp hp 2 kγσn(u − v )k 1 ≤ cγ ku − v k 3 H 2 (ΓC ) H 2 (ΓC ) 2 hp hp 2 ≤ cγ0 ku − v k 1 H 2 (Σ) 1+2ν 2 h 2 2 hp 2 ≤ cγ0 1+2ν kuk ˜ 1+ν +cγ0 ku − u k 1 (3.84) p H (Σ) H˜ 2 (Σ) and with the continuity of Ehp (Lemma2.6)

hp hp 2 hp 2 kEn (u − u)k − 1 ≤ CEhp ku − uk 1 (3.85) 2 H 2 (ΓC ) 2 H 2 (Σ) From Lemma2.6 and Lemma2.11, we have

1+2ν 1+2ν h 2 h 2 h 2 kE (u)k ≤ kψk ν ≤ C kuk (3.86) n − 1 1+2ν H˜ (Γ) 1+2ν ˜ 1+ν H 2 (ΓC ) p p H (Σ) Using (3.84)-(3.86), we obtain

1+2ν 0 h 2 2 1 1 hp 2 A7,n ≤ C kuk 1+ν +[cγ0 ( + ) + CEhp ]ku − uk 1 (3.87) p1+2ν H˜ (Σ) 2 2 2 H 2 (Σ) Similarly, we obtain Z t hp hp hp A7,t = inf γE (u )σt(u − v ) ds hp hp v ∈Vhp ΓC 1+2ν 0 h 2 2 1 1 hp 2 ≤ C kuk 1+ν +[cγ0 ( + ) + CEhp ]ku − uk 1 p1+2ν H˜ (Σ) 2 2 2 H 2 (Σ) Z hp hp h hp hp A8,n = inf γE (u )E (u − v ) ds hp n n v ∈Vhp ΓC 1+2ν 0 h 2 2 1 1 hp 2 ≤ C kuk 1+ν +[cγ0 ( + ) + CEhp ]ku − uk 1 p1+2ν H˜ (Σ) 2 2 2 H 2 (Σ) Z hp hp h hp hp A8,t = inf γE (u )E (u − v ) ds hp t t v ∈Vhp ΓC 1+2ν 0 h 2 2 1 1 hp 2 ≤ C kuk 1+ν +[cγ0 ( + ) + CEhp ]ku − uk 1 p1+2ν H˜ (Σ) 2 2 2 H 2 (Σ) (3.88)

We estimate now the term Z Hq Hq n hp A9,n = inf γ(λ − µ )E (u ) ds (3.89) Hq n n hp µn ∈Mn,Hq ΓC We have Z Z Hq n hp Hq n hp A9,n ≤ γ(λn − λn)Eh (u ) ds + γ(λn − µn )Eh (u ) ds (3.90) ΓC ΓC

60 3.3 A priori error analysis for frictional contact problem

We have to estimate the ﬁrst term in (3.90) Z Hq n hp A9,1,n = γ(λn − λn)Eh (u ) ds (3.91) ΓC Z Z Hq n hp hp Hq n hp A9,1,n = γ(λn − λn)Eh (u − v ) ds − γ(λn − λn)Eh (u − v ) ds ΓC ΓC Z Hq n + γ(λn − λn)Eh (u) ds (3.92) ΓC Employing Cauchy Schwarz and Young’s inequality, we have

3 1 Hq 2 γ0h n hp hp 2 γ0h n hp 2 A9,1,n ≤ kγ 2 (λn − λ )k + kE (u − v )k + kE (u − v )k 2 n L2(ΓC ) p2 h L2(ΓC ) p2 h L2(ΓC ) γ h + 0 kEnuk2 (3.93) p2 h L2(ΓC ) with the continuity of Ehp, we have n 2 2 kE uk ≤ αkuk 1+ν , for ν = 0, 1 (3.94) h L2(ΓC ) H˜ (Σ) and

1 n hp 2 1 hp 2 kγ 2 E (u − v )k ≤ αkγ 2 (u − v )k 1 h L2(ΓC ) H (Γ) h = αγ ku − I uk2 0 p2 hp H1(Γ) 1+2ν h 2 = Cαγ0 kuk 1+ν . (3.95) p2+2ν H˜ (Σ) Using (3.78), (3.94) and (3.95), we obtain 1+2ν 1+2ν 3 1 Hq 2 γ0 h 2 γ0 h 2 A9,1,n ≤ kγ 2 (λn − λ )k + kuk 1+ν + kuk 1+ν 2 n L2(ΓC ) p2+2ν H˜ (Σ) p1+2ν H˜ (Σ)

γ0 hp 2 γ0 h 2 + ku − uk 1 +α kuk 1+ν (3.96) H 2 (Σ) p2 H˜ (Σ) We estimate now the second term in (3.90) Z Hq n hp A9,2,n = γ(λn − µn )Eh (u ) ds (3.97) ΓC n The continuity of Eh and Lemma3.6, we get

h n hp 2 h hp 2 hp 2 2 kEh u kL (Γ )≤ α ku k 1 ≤ αku k 1 ≤ αkuk 1+ν (3.98) p2 2 C p2 H (Σ) H 2 (Σ) H˜ (Σ) Using Cauchy Schwarz and (3.98), we obtain

1 1 h 2 Hq h 2 n hp 2 A9,2,n ≤ γ0 kλn − µ k kE u k p n L2(ΓC ) p h L2(ΓC ) 1 ν h 2 H ≤ γ kλ k ν kuk 1+ν 0 p qν n H (ΓC ) H˜ (Σ) 1 ν h 2 H 2 ≤ γ0 kuk 1+ν (3.99) p qν H˜ (Σ)

61 3 Stabilized mixed hp-BEM in Linear Elasticity

Finally we obtain

1+2ν 1+2ν 3 1 Hq 2 γ0 h 2 γ0 h 2 A9,n ≤ kγ 2 (λn − λ )k + kuk 1+ν + kuk 1+ν 2 n L2(ΓC ) p2+2ν H˜ (Σ) p1+2ν H˜ (Σ) 1 ν γ0 hp 2 γ0 h 2 h 2 H 2 + ku − uk 1 +α kuk 1+ν +γ0 kuk 1+ν (3.100) H 2 (Σ) p2 H˜ (Σ) p qν H˜ (Σ)

Similarly to (3.100), we obtain Z Hq Hq t hp A9,t = inf γ(λ − µ )E (u ) ds Hq t t h µt ∈Mt,Hq ΓC 1+2ν 1+2ν 3 1 Hq 2 γ0 h 2 γ0 h 2 ≤ kγ 2 (λt − λ )k + kuk 1+ν + kuk 1+ν 2 t L2(ΓC ) p2+2ν H˜ (Σ) p1+2ν H˜ (Σ) 1 ν γ0 hp 2 γ0 h 2 h 2 H 2 + ku − uk 1 +α kuk 1+ν +γ0 kuk 1+ν (3.101) H 2 (Σ) p2 H˜ (Σ) p qν H˜ (Σ)

1 Hq h 2 2 2 Note that γ0 is small enough. Moring the terms γ0ku−u k 1 , kγ (λn−λn )kL (Γ ) H˜ 2 (Σ) 2 C 1 Hq 2 and kγ 2 (λt − λ )k to the left hand side, we obtain the a priori error estimate t L2(ΓC ) of the theorem.

We consider the terms Z Hq A1,n = inf (λn − µn)un ds (3.102) µ ∈M n n ΓC and Z Hq A1,t = inf (λt − µt)ut ds (3.103) µ ∈M (F) n t ΓC

Remark 3.2. The estimation of the terms A1,n and A1,t seems to be problematic, due to the nonconformity of our approach. Here the positivity condition is enforced only on the discrete set of the Gauss Lobatto points.

1 − 1 Remark 3.3. The estimate in Theorem3.3 is of order h 4 p 4 (when H = h and p = q). In the nonstabilized case (see.Chapter2), for vanishing gap function g = 0, we obtain 1 − 1 a convergence rate of order h 4 p 4 similar to the stabilized case, when we assume the inf-sup condiction to hold.

For the completeness of the convergence analysis we also consider the h-version for p = 1 and q = 1.

In order to estimate the terms A1,n and A1,t, we have to deﬁne the space of the discrete Lagrange multiplier introduced in [36, 37, 45] for the FEM.

H H Mn,H := {λn ∈ WH : λn ≥ 0 on ΓC }

62 3.3 A priori error analysis for frictional contact problem

H H Mt,H := {λt ∈ WH : |λt | ≤ F on ΓC } where H H WH := {µ ∈ C(ΓC ): ∀J ∈ TH , µ |J ∈ P1(J)}

It is a conforming discretization on multiplier as Mn,H ⊂ Mn and Mt,H ⊂ Mt.

We obtain the following a priori error estimate which proposes a convergence rate of 1 O(h 4 ) (when H = h)

Corollary 3.3. Let (u, λ) ∈ V × M be the solution of the problem (3.4) and (uh, λH ) 1+ν ν the solution of the discrete problem (3.8). Assume that u ∈ H˜ (Σ), λ ∈ H (ΓC ) for 1 some ν ∈ [0, ]. Suppose that kλ k ν +kλ k ν +kFk ≤ kuk 1+ν . Then 2 n H (ΓC ) t H (ΓC ) L2(ΓC ) H˜ (Σ) there exists a constant C > 0 independent of h and H, such that

1 1 h h 2 H 2 H ku − u k 1 +kψ − ψ k − 1 +kγ (λn − λn )kL (Γ )+kγ (λt − λt )kL (Γ ) H˜ 2 (Σ) H˜ 2 (Γ) 2 C 2 C 1 ν+ 1 1 ν ≤ C h 4 + H 2 + h 2 H kuk 1+ν H˜ (Σ) (3.104)

Proof. It is suﬃcient to estimate the terms A1,n and A1,t .

We have to estimate Z H A1,n = inf (λn − µn)un ds (3.105) µ ∈M n n ΓC

H Setting µn = 0, since un ≤ 0 and λn ≥ 0 on ΓC , we have Z Z H H A1,n = inf (λn − µn)un ds ≤ λn un ds ≤ 0 (3.106) µ ∈M n n ΓC ΓC

We now estimate the term Z H A1,t = inf (λt − µt)ut ds (3.107) µ ∈M (F) n t ΓC

We consider the Lagrange interpolation operator Ih deﬁned on the Gauss-Lobatto points mapping onto Vh, where Vh is the space of continuous piecewise polynomials for the discretization of u for p = 1.

We have Z Z H λt Ihut ds ≤ F |Ihut| ds (3.108) ΓC ΓC

63 3 Stabilized mixed hp-BEM in Linear Elasticity

choosing µt = λt and use the condion λtut − F|ut| = 0 we get Z H (λt − λt)ut ds = ΓC

Z Z Z Z H H (λt − λt)(ut − Ihut) ds + (λt − λt)Ihut ds + λtut ds − F |ut| ds ΓC ΓC ΓC ΓC

Z Z Z H ≤ (λt − λt)(ut − Ihut) ds + λt(ut − Ihut) ds + F (|Ihut|−|ut|) ds ΓC ΓC ΓC

Z Z Z H ≤ (λt − λt)(ut − Ihut) ds + λt(ut − Ihut) ds + F (|Ihut − ut|) ds ΓC ΓC ΓC (3.109) and we obtain Z H (λt − λt)ut ds ΓC 1 1 2 H − 2 ≤ kγ (λt − λt )kL2(ΓC )γ kut − IhutkL2(ΓC )+kλtkL2(ΓC )kut − IhutkL2(ΓC )

+ kFkL2(ΓC )kut − IhutkL2(ΓC )

1 H 2 C 2ν+1 2 ν+1 2 ≤ kγ 2 (λ − λ )k + h kuk ν+1 +Ch kuk ν+1 t t L2(Γ ) ˜ ˜ C γ0 H (Σ) H (Σ) (3.110)

The corollary is established by employing the estimation of the terms in Theorem3.3.

3.4 Reliable and eﬃcient a posteriori error estimates for stabilized hp-BEM for frictional contact problems

3.4.1 Reliability of the BEM a posteriori error estimate

Let (u, λ) be the solution of the continuous problem (3.4) and (uhp, λHq) the solution of the discrete problem (3.8). We now derive an upper bound for |ku − uhpk|2, where

hp 2 hp 2 hp 2 Hq 2 |ku − u k| := ku − u k 1 +kψ − ψ k 1 +kλ − λ k 1 (3.111) ˜ 2 − 2 ˜ − 2 H (Σ) H (Σ) H (ΓC )

64 3.4 Reliable and eﬃcient a posteriori error estimates for stabilized hp-BEM for frictional contact problems

1 ˜ 2 Lemma 3.7. ([19].Lemma 3.2.9). There exists an operator Πhp : H (Σ) → Vhp, which 1 2 is stable in the H˜ -norm and has the quasioptimal approximation proprietes in the L2- 1 norm,ie. there exists a constant C , independent of h and p such that for all u ∈ H˜ 2 (Σ) there holds

kΠhpuk 1 ≤ Ckuk 1 (3.112) H˜ 2 (Σ) H˜ 2 (Σ) (1−) h 2 ku − ΠhpukL (Σ) ≤ C kuk 1 (3.113) 2 p H˜ 2 (Σ)

1 with arbitrary small ∈ (0; 2 )

Theorem 3.4. Let (u, λ) be the solution of the problem (3.4) and (uhp, λHq) the solution of the discrete problem (3.8). Then there holds the estimate for 1 > 0 arbitrary:

hp 2 hp 2 X 2 Hq + hp + ku − u k 1 +kψ − ψ k − 1 . ηh(I) + h(λn ) , (g − un ) iΓC H˜ 2 (Σ) H 2 (Σ) I∈Thp 1 + k(λHq)−k2 + k(uhp − g)+k2 n ˜ − 1 n 1 H 2 (ΓC ) 41 H 2 (ΓC ) Hq 2 Hq + 2 + 1kλ − λ k 1 +k(|λt | − F) k 1 ˜ − 2 ˜ − 2 H (ΓC ) H (ΓC ) Z Hq − hp Hq hp − + |(|λt | − F) ||ut | + 2(λt ut ) ds ΓC where for an arbitrary segment I

h 1− h 1− η2(I) = I kt − S uhpk2 + I k(−λHq) − S uhpk2 h hp L2(I∩ΓN ) hp L2(I∩ΓC ) pI pI h h + I kλHq + σh(uhp)k2 + I kλHq + σh(uhp)k2 2 n n L2(I∩ΓC ) 2 t t L2(I∩ΓC ) pI pI ∂ ∗ hp 2 + hI k (V (ψ − ψ ))k . ∂s hp L2(I)

Proof. Since hS·, ·i, hV ·, ·i are positive deﬁnite, there exist constants cs, cv > 0 such that

hp 2 hp hp hp csku − u k 1 +cvkψ − ψ k − 1 ≤ hSu − Shpu , u − u i H˜ 2 (Σ) H 2 (Σ) ∗ hp hp + hV (ψhp − ψ ), ψ − ψ i. (3.114)

65 3 Stabilized mixed hp-BEM in Linear Elasticity

The Galerkin orthogonality property (Lemma 3.4) shows that

hp 2 hp hp hp csku − u k 1 +cvkψ − ψ k 1 ≤ hSu − Shpu , u − u iΣ H˜ 2 (Σ) H 2 (Σ) ∗ hp hp hp hp hp Hq hp hp + hV (ψp − ψ ), ψ − ψ iΣ + hSu − Shpu , u − v iΣ + b(λ − λ , u − v ) Z Z Hq h hp h hp hp Hq h hp h hp hp + γ(λn + σn(u ))σn(u − v ) ds + γ(λt + σt (u ))σt (u − v ) ds ΓC ΓC hp hp ∗ hp hp Hq hp hp ≤ hSu − Shpu , u − v i + hV (ψhp − ψ ), ψ − ψ i + b(λ − λ , u − v ) Z Z Hq h hp h hp hp Hq h hp h hp hp + γ(λn + σn(u ))σn(u − v ) ds + γ(λt + σt (u ))σt (u − v ) ds ΓC ΓC hp hp hp hp ≤ ht, u − v iΓN − b(λ, u − v ) − hShpu , u − v i ∗ hp hp Hq hp hp + hV (ψhp − ψ ), ψ − ψ i + b(λ − λ , u − v ) Z Z Hq h hp h hp hp Hq h hp h hp hp + γ(λn + σn(u ))σn(u − v ) ds + γ(λt + σt (u ))σt (u − v ) ds ΓC ΓC hp hp Hq hp hp ≤ ht − Shpu , u − v iΓN + h(−λ ) − Shpu , u − v iΓC Hq hp ∗ hp hp + b(λ − λ , u − u) + hV (ψhp − ψ ), ψ − ψ i Z Z Hq h hp h hp hp Hq h hp h hp hp + γ(λn + σn(u ))σn(u − v ) ds + γ(λt + σt (u ))σt (u − v ) ds ΓC ΓC (3.115)

We estimate the ﬁrst and the second terms, employing the Cauchy-Schwarz inequality:

hp hp X hp hp A = ht − Shpu , u − v iΓN ≤ kt − Shpu kL2(I)ku − v kL2(I) (3.116) I∈Thp∩ΓN Hq hp hp X Hq hp hp B = h(−λ ) − Shpu , u − v iΓC ≤ k(−λ ) − Shpu kL2(I)ku − v kL2(I) I∈Thp∩ΓC (3.117)

We apply a result from [15], we obtain

∗ hp hp ∗ hp hp D := hV (ψ − ψ ), ψ − ψ i ≤ kV (ψ − ψ )k 1 kψ − ψk 1 hp hp H 2 (Σ) H− 2 (Σ) 1   2 X ∂ ∗ hp 2 hp ≤ c  hI k (V (ψhp − ψ ))k  kψ − ψk − 1 . (3.118) ∂s L2(I) H 2 (Σ) I∈Thp

hp hp hp Using lemma 3.7, we choose v = u + Πhp(u − u ) to obtain the following estimate:

(1−) 2 hp hI hp ku − v kL2(I)≤ C ku − u k 1 . (3.119) pI H 2 (ω(I))

Here ω(I) is a neighbourhood of I.

66 3.4 Reliable and eﬃcient a posteriori error estimates for stabilized hp-BEM for frictional contact problems

The continuity of Shp and Young’s inequality imply

1 1 −1 2 h hp hp −1 2 h hp p h kσn(v − u )kL2(Γ) = p h kσn(Πhp(u − u ))kL2(Γ) −1 1 ˜ hp ≤ p h 2 CkΠhp(u − u )kH1(Γ) hp ≤ C˜kΠhp(u − u )k 1 H 2 (Γ) hp ≤ CC˜ ku − u k 1 (3.120) H 2 (Γ)

hp hp hp Since v = u + Πhp(u − u ), we get

Z Hq h hp h hp hp En = γ(λn + σn(u ))σn(u − v ) ds ΓC Z X Hq h hp h hp = γ(λn + σn(u ))σn(Πhp(u − u)) ds I I∈ΓC  1   1  Z 2 2 X hI Hq h hp hI h hp ≤ γ0   (λn + σn(u ))   σn(Πhp(u − u )) ds I pI pI I∈ΓC 1 1   2   2 X hI X hI ≤ γ kλHq + σh(uhp)k2 kσh(Π (u − uhp))k2 0  2 n n L2(I)  2 n hp L2(I) pI pI I∈ΓC I∈ΓC 1   2 X hI Hq h hp 2 hp ≤ Cˆ kλ + σ (u )k ku − u k 1 . (3.121)  2 n n L2(I) H 2 (Γ) pI I∈ΓC

Similarly to (3.121), we obtain

Z Hq h hp h hp hp Et = γ(λt + σt (u ))σt (u − v ) ds ΓC 1   2 X hI Hq h hp 2 hp ≤ Cˆ kλ + σ (u )k ku − u k 1 . (3.122)  2 t t L2(I) H 2 (Γ) pI I∈ΓC

Finally we estimate the term

Hq hp Hq hp Hq hp C = b(λ − λ , u − u) = hλn − λn , un − uni + hλt − λt , ut − uti. (3.123)

Hq + + Using the condition hλn, un − gi = 0 and h(λn ) , un − gi ≤ 0 where v = max{0, v}

67 3 Stabilized mixed hp-BEM in Linear Elasticity and v− = min{0, v}, i.e. v = v+ + v−.

Hq hp Hq + hp Hq − hp hλn − λn , un − uni = hλn − (λn ) , un − uni − h(λn ) , un − uni Hq + hp Hq − hp = hλn − (λn ) , un − g + g − uni − h(λn ) , un − uni Hq + hp = hλn − (λn ) , un − gi + hλn, g − uni Hq + Hq − hp − h(λn ) , g − uni − h(λn ) , un − uni Hq + hp Hq − hp ≤ hλn − (λn ) , un − gi − h(λn ) , un − uni Hq + hp hp + hp − = h(λn ) , g − un i − hλn, (g − un ) + (g − un ) i Hq − hp − h(λn ) , un − uni Hq + hp Hq Hq + Hq − hp − ≤ h(λn ) , g − un i + hλn − λn − (λn ) − (λn ) , (g − un ) i Hq − hp − h(λn ) , un − uni Hq + hp + Hq hp − = h(λn ) , (g − un ) i + hλn − λn, (g − un ) i Hq − hp − Hq − hp − h(λn ) , (g − un ) i − h(λn ) , un − uni Hq + hp + Hq hp − = h(λn ) , (g − un ) i + hλn − λn, (g − un ) i Hq − hp − h(λn ) , un − uni Hq − hp ≤ k(λn ) k − 1 kun − unk 1 H˜ 2 (ΓC ) H˜ 2 (Σ) Hq hp − Hq + hp + + kλn − λnk − 1 k(g − un ) k 1 +h(λn ) , (g − un ) iΓC . H˜ 2 (ΓC ) H 2 (ΓC ) (3.124)

By the contact condition, −λtut + F|ut| = 0. Setting λt = ξF with |ξ| ≤ 1

Hq hp Hq hp Hq hp hλt − λt , ut − uti = −λtut + λt ut + λtut − λt ut Hq hp Hq hp = −F|ut| + λt ut + ξFut − λt ut Hq hp Hq hp ≤ −F|ut| + λt ut + F|ut | − λt ut Hq + hp Hq hp ≤ (|λt | − F) |ut| + F|ut | − λt ut Hq + hp Hq + hp hp Hq hp ≤ (|λt | − F) |ut − ut | + (|λt | − F) |ut | + F|ut | − λt ut Hq + hp ≤ k(|λt | − F) k − 1 kut − ut k 1 H˜ 2 (ΓC ) H 2 (ΓC ) Hq + Hq hp Hq hp Hq hp + [(|λt | − F) − (|λt | − F)]|ut | − λt ut + |λt ||ut | Hq + hp ≤ k(|λt | − F) k − 1 kut − ut k 1 H˜ 2 (ΓC ) H 2 (ΓC ) Hq − hp Hq hp − + |(|λt | − F) ||ut | + 2(λt ut ) . (3.125)

hp Using Young’s inequality we move the term ku − u k 1 to the left hand side. The H˜ 2 (Σ) estimate of the theorem follows.

Now we ﬁnd an upper bound on the discretization error λ − λHq.

Lemma 3.8. Let λ solve the saddle point problem (3.4), and λHq the solution of the

68 3.4 Reliable and eﬃcient a posteriori error estimates for stabilized hp-BEM for frictional contact problems discrete problem (3.8). Then there holds Hq 2 0 hp 2 hp 2 00 X 2 kλ − λ k − 1 ≤ C ku − u k 1 +kψ − ψ k − 1 + C ξh(I). H 2 (ΓC ) H˜ 2 (Σ) H˜ 2 (Γ) I⊂Thp (3.126) where h 1− h 1− ξ2(I) = I kt − S uhpk2 + I k(−λHq) − S uhpk2 h hp L2(I∩ΓN ) hp L2(I∩ΓC ) pI pI h h + I kλHq + σh(uhp)k2 + I kλHq + σh(uhp)k2 2 n n L2(I∩ΓC ) 2 t t L2(I∩ΓC ) pI pI (3.127)

hp Proof. Let v ∈ V and v := Πhpv ∈ Vhp we have hλ − λHq, vi = hλ − λHq, v − vhpi + hλ − λHq, vhpi

Using Galerkin orthogonality and the formulation (3.4) we obtain

Hq Hq hp hp hp hλ − λ , vi = hλ − λ , v − v i − hSu − Shpu , v iΣ Z Z Hq h hp h hp Hq h hp h hp − γ(λn + σn(u ))σn(v ) ds − γ(λt + σt (u ))σt (v ) ds ΓC ΓC = ht, v − vhpi − hSu, v − vhpi − hλHq, v − vhpi hp hp hp − hSu − Shpu , viΣ − hSu − Shpu , v − viΣ Z Z Hq h hp h hp Hq h hp h hp − γ(λn + σn(u ))σn(v ) ds − γ(λt + σt (u ))σt (v ) ds ΓC ΓC hp hp H hp hp = ht − Shpu , v − v iΓN + h(−λ ) − Shpu , v − v iΓC hp − hSu − Shpu , viΣ Z Z Hq h hp h hp Hq h hp h hp − γ(λn + σn(u ))σn(v ) ds − γ(λt + σt (u ))σt (v ) ds ΓC ΓC

We estimate the terms on the right hand side:

hp hp X hp hp A = ht − Shpu , v − v iΓN ≤ kt − Shpu kL2(I)kv − v kL2(I) (3.128) I∈Thp∩ΓN Hq hp hp X Hq hp hp B = h(−λ ) − Shpu , v − v iΓC ≤ k(−λ ) − Shpu kL2(I)kv − v kL2(I) I∈Thp∩ΓC (3.129)

hp hp C = hSu − Shpu , viΣ ≤ (Cs + CE )ku − u k 1 kvk 1 hp H˜ 2 (Σ) H˜ 2 (Σ) hp + C0kψ − ψ k − 1 kvk 1 . (3.130) H˜ 2 (Γ) H˜ 2 (Σ)

69 3 Stabilized mixed hp-BEM in Linear Elasticity

Using the inverse inequality as well as Lemma 3.7:

Z Z Hq h hp h hp X Hq h hp h Dn = γ(λn + σn(u ))σn(v ) ds ≤ γ(λn + σn(u ))σn(Πhpv) ΓC I I∈Thp  1   1  Z 2 2 X hI Hq h hp hI h = γ0   (λn + σn(u ))   σn(Πhpv) I pI pI I∈Thp 1   2 X hI Hq h hp 2 ≤ Cˆ kλ + σ (u )k kvk 1 . (3.131)  2 n n L2(I) H 2 (Γ) pI I∈ΓC

Similarly to (3.131), we get

  1 Z 2 Hq h hp h hp X hI Hq h hp 2 Dt = γ(λ + σ (u ))σ (v ) ds ≤ Cˆ kλ + σ (u )k kvk 1 t t t  2 t t L2(I) H 2 (Γ) ΓC pI I∈ΓC (3.132)

hp Recalling v = Πhpv in A and B and Lemma 3.7 we obtain

Hq hp hp hλ − λ , vi ≤ (Cs + CE )ku − u k 1 +C0kψ − ψ k − 1 kvk 1 hp H˜ 2 (Σ) H˜ 2 (Γ) H˜ 2 (Σ)

 1−  2 X hI hp 1 + C  kt − Shpu kL2(I) kvk pI H˜ 2 (Σ) I∈Thp∩ΓN

 1−  2 X hI Hq hp 1 + C  k(−λ ) − Shpu kL2(I) kvk pI H˜ 2 (Σ) I∈Thp∩ΓN  1  2 X hI Hq h hp ˆ 1 + C  kλn + σn(u )kL2(I) kvk pI H 2 (Γ) I∈ΓC  1  2 X hI Hq h hp ˆ 1 + C  kλt + σt (u )kL2(I) kvk pI H 2 (Γ) I∈ΓC (3.133)

By deﬁnition of the dual norm and (a+b)2 ≤ 2a2 +2b2, the assertion (3.126) follows.

Theorem 3.5. Let (u, λ) be the solution of problem (3.4) and (uhp, λHq) the solution

70 3.4 Reliable and eﬃcient a posteriori error estimates for stabilized hp-BEM for frictional contact problems of the discrete problem (3.8). Then the following a posteriori estimate holds:

hp 2 hp 2 Hq 2 ku − u k 1 +kψ − ψ k 1 + kλ − λ k 1 ˜ 2 − 2 ˜ − 2 . H (Σ) H (Σ) H (ΓC ) X 2 Hq + hp + ηh(I) + h(λn ) , (g − un ) iΓC I∈Thp Hq − 2 hp + 2 + k(λn ) k − 1 +k(un − g) k 1 H˜ 2 (ΓC ) H 2 (ΓC ) Hq + 2 + k(|λt | − F) k − 1 H˜ 2 (ΓC ) Z Hq − hp Hq hp − + |(|λt | − F) ||ut | + 2(λt ut ) ds ΓC (3.134) where

h 1− h 1− η2(I) = I kt − S uhpk2 + I k(−λHq) − S uhpk2 h hp L2(I∩ΓN ) hp L2(I∩ΓC ) pI pI h h + I kλHq + σh(uhp)k2 + I kλHq + σh(uhp)k2 2 n n L2(I∩ΓC ) 2 t t L2(I∩ΓC ) pI pI ∂ ∗ hp 2 + hI k (V (ψ − ψ ))k ∂s hp L2(I)

Proof. The estimate follows immediately from Lemma 3.8 and Theorem 3.4

3.4.2 Eﬃciency of the BEM a posteriori error estimate

In this section we derive an eﬃciency of the a posteriori error estimates. We follow ideas of [15], [16], [19].

Assumption 3.1. There exists a constant C > 0 such that

|I| pI ∀I,J ∈ Thp < C, < C (3.135) |J| pJ and C is independent of I,J . Let

hmax := max |I|, hmin := min |I| I∈Thp I∈Thp

pmax := max pI , pmin := min pI I∈Thp I∈Thp

For the simplicity of the the presentation, we assume that the gap function g = 0.

Lemma 3.9. There exists a constant c > 0 such that for any element I ∈ Thp the local

71 3 Stabilized mixed hp-BEM in Linear Elasticity

error indicator ηh(I) satisﬁes

h h 0 1 cγ η2(I) ≤ I kW (u − uhp)k2 + I k(K + )(ψ − ψhp)k2 0 h L2(I∩Σ) L2(I∩Σ) pI pI 2 ∂ hp 2 ∂ 1 hp 2 + hI k (V (ψ − ψ ))k +hI k (K + )(u − u )k ∂s L2(I) ∂s 2 L2(I) 1 Hq 2 1 hp 2 + p kγ 2 (λ − λ )k +kγ 2 σ (u − u )k I L2(I∩ΓC ) n L2(I∩ΓC ) 1 hp 2 + kγ 2 σ (u − u )k (3.136) t L2(I∩ΓC )

hI the stabilization parameter γ is deﬁned on each element I as γ = γ0 2 , where γ0 > 0 is pI independent of h and p and γ0 is chosen small enough, where

h h η2 = I kt − S uhpk2 + I k(−λHq) − S uhpk2 h hp L2(I∩ΓN ) hp L2(I∩ΓC ) pI pI h h + I kλHq + σ (uhp)k2 + I kλHq + σ (uhp)k2 2 n n L2(I∩ΓC ) 2 t t L2(I∩ΓC ) pI pI ∂ h h + h k (V (ψ∗ − ψhp))k2 + I kEh(uhp)k2 + I kEh(uhp)k2 I hp L2(I) 2 n L2(I∩ΓC ) 2 t L2(I∩ΓC ) ∂s pI pI

Proof. Noting that t = Su|ΓN for the exact solution u , we obtain for I ⊆ ΓN :

h γ I kt − S uhpk2 = γp kSu − S uhk2 0 hp L2(I∩ΓN ) I hp L2(I∩ΓN ) pI hp 0 1 hp 2 = γpI kW (u − u ) + (K + )(ψ − ψ )k 2 L2(I∩ΓN ) ≤ 2γp kW (u − uhp)k2 I L2(I∩ΓN ) 0 1 hp 2 + 2γpI k(K + )(ψ − ψ )k . 2 L2(I∩ΓN )

Note that λ = −Su|ΓC . Then we obtain for I ⊆ ΓC

h γ I k(−λHq) − S uhpk2 ≤ 2γp kλ − λHqk2 +kSu − S uhpk2 0 hp L2(I) I L2(I) hp L2(I) pI ≤ 4γp kW (u − uhp)k2 I L2(I) 0 1 hp 2 + 4γpI k(K + )(ψ − ψ )k 2 L2(I) 1 Hq 2 + 2p kγ 2 (λ − λ )k I L2(I) and for any I ⊆ Γ we have

∂ hp ∗ 2 ∂ hp 1 hp 2 γ0hI k (V (ψ − ψ ))k = γ0hI k (V (ψ − (K + )u ))k ∂s hp L2(I) ∂s 2 L2(I) ∂ hp 2 ≤ 2γ0hI k (V (ψ − ψ ))k ∂s L2(I) ∂ 1 hp 2 + 2γ0hI k (K + )(u − u )k . ∂s 2 L2(I)

72 3.4 Reliable and eﬃcient a posteriori error estimates for stabilized hp-BEM for frictional contact problems

hI Hq hp 2 1 Hq 2 1 hp 2 γ kλ + σ (u )k = kγ 2 (λ − λ )k +kγ 2 σ (u − u )k 0 2 n n L2(I∩ΓC ) n n L2(I) n L2(I∩ΓC ) pI 1 Hq 2 1 hp 2 ≤ kγ 2 (λ − λ )k +kγ 2 σ (u − u )k L2(I) n L2(I∩ΓC ) (3.137)

Similarly to (3.137), we get

hI Hq hp 2 1 Hq 2 1 hp 2 γ kλ + σ (u )k ≤ kγ 2 (λ − λ )k +kγ 2 σ (u − u )k 0 2 t t L2(I∩ΓC ) L2(I) t L2(I∩ΓC ) pI (3.138)

h h γ I kEh(uhp)k2 = γ I k(S − S )uhp · nk2 0 2 n L2(I∩ΓC ) 0 2 hp L2(I∩ΓC ) pI pI h h ≤ γ I kSu − S uhpk2 +γ I kS(u − uhp) · nk2 0 2 hp L2(I∩ΓC ) 0 2 L2(I∩ΓC ) pI pI h 0 1 ≤ γ I kW (u − uhp)k2 +k(K + )(ψ − ψhp)k2 0 2 L2(I∩ΓC ) L2(I∩ΓC ) pI 2 1 hp 2 + kγ 2 σ (u − u )k (3.139) n L2(I∩ΓC )

Finally we have h h 0 1 γ I kEh(uhp)k2 ≤ γ I kW (u − uhp)k2 +k(K + )(ψ − ψhp)k2 0 2 t L2(I∩ΓC ) 0 2 L2(I∩ΓC ) L2(I∩ΓC ) pI pI 2 1 hp 2 + kγ 2 σ (u − u )k . (3.140) t L2(I∩ΓC )

Lemma 3.10. Let Php be the L2(Γ)-projection operator onto Whp, i.e Php : L2(Γ) −→ Whp such that

hPhpψ − ψ, Φi = 0, ∀Φ ∈ Whp (3.141) and for any real numbers µ ≥ 0 there exists a constant C such that

µ µ h ∀ψ ∈ H (Γ) kψ − P ψk ≤ C kψk µ . (3.142) hp L2(Γ) p H (Γ)

Then there holds 2 h 2 kψ − Phpψk − 1 ≤ C kψkL (Γ). (3.143) H 2 (Γ) p 2

In particular 2 h 2 kψ − Phpψk − 1 ≤ C kψ − PhpψkL (Γ). (3.144) H 2 (Γ) p 2

73 3 Stabilized mixed hp-BEM in Linear Elasticity

Proof. We observe that

hψ − Phpψ, φi kψ − Phpψk − 1 = sup H 2 (Γ) 1 kφk 1 φ∈H 2 (Γ) H 2 (Γ) hψ − P ψ, φ − P φi = sup hp hp 1 kφk 1 φ∈H 2 (Γ) H 2 (Γ) hψ, φ − P φi = sup hp 1 kφk 1 φ∈H 2 (Γ) H 2 (Γ)

kφ − PhpφkL2(Γ) ≤ kψkL2(Γ) sup 1 kφk 1 φ∈H 2 (Γ) H 2 (Γ) 1 h 2 ≤ C kψk (3.145) p L2(Γ) and

hψ − Phpψ, φi kψ − Phpψk − 1 = sup H 2 (Γ) 1 kφk 1 φ∈H 2 (Γ) H 2 (Γ) hψ − P ψ, φ − P φi = sup hp hp 1 kφk 1 φ∈H 2 (Γ) H 2 (Γ)

kψ − Phpψk kφ − Phpφk ≤ sup L2(Γ) L2(Γ) 1 kφk 1 φ∈H 2 (Γ) H 2 (Γ) 1 h 2 = C kψ − P ψk (3.146) p hp L2(Γ)

Theorem 3.6. Let (u, λ) be the solution of problem (3.4) and (uhp, λHq) the solution of the discrete problem (3.8). Assume that λ ∈ L2(ΓC ). Then there exists a constant c(γ0) independent of h, p, H and q such that X 1 2 2 hp 2 hp 2 2 Hq 2 c(γ0) ηh(I) ≤ Cpmax ku − u k 1 +kψ − ψ k − 1 +kγ (λ − λ )kL (Γ ) H 2 (Σ) H 2 (Γ) 2 C I∈Thp 2 2 2 + hmax kuk 3 +kψk 1 (3.147) H˜ 2 (Σ) H 2 (Γ) where h h η2(I) = I kt − S uhpk2 + I k(−λHq) − S uhpk2 h hp L2(I∩ΓN ) hp L2(I∩ΓC ) pI pI h h + I kλHq + σ (uhp)k2 + I kλHq + σ (uhp)k2 2 n n L2(I∩ΓC ) 2 t t L2(I∩ΓC ) pI pI ∂ h h + h k (V (ψ∗ − ψhp))k2 + I kEh(uhp)k2 + I kEh(uhp)k2 I hp L2(I) 2 n L2(I∩ΓC ) 2 t L2(I∩ΓC ) ∂s pI pI

74 3.4 Reliable and eﬃcient a posteriori error estimates for stabilized hp-BEM for frictional contact problems and we have Z Hq + hp + Hq hp (λn ) (un ) ds ≤ kλ − λ kL2(ΓC )ku − u kL2(ΓC ) ΓC

hp + kλkL2(ΓC )ku − u kL2(ΓC )

Hq − 2 Hq 2 k(λ ) k 1 ≤ kλ − λ k n − L2(ΓC ) H˜ 2 (ΓC )

Z Hq − hp Hq hp − Hq hp |(|λt | − F) ||ut | + 2(λt ut ) ds . kλ − λ kL2(ΓC )ku − u kL2(ΓC ) ΓC

Hq + kλ − λ kL2(ΓC )kukL2(ΓC )

hp + kλkL2(ΓC )ku − u kL2(ΓC )

Hq + 2 Hq k(|λt | − F) k − 1 ≤ kλ − λ kL2(ΓC ) H˜ 2 (ΓC )

2 hp + 2 pmax hp k(un ) k 1 ≤ ku − u kL2(ΓC ). (3.148) H 2 (ΓC ) hmin

Proof. Using the continuity of W and K and Lemma3.9 we obtain

hmax hp 2 hmax hp 2 γ kW (u − u )k ≤ Cγ ku − u k 1 (3.149) 0 L2(Σ) 0 H (Γ) pmin pmin

∂ 1 hp 2 1 hp 2 γ0hmaxk (K + )(u − u )k ≤ γ0hmaxk(K + )(u − u )k 1 ∂s 2 L2(Γ) 2 H (Γ)

hp 2 ≤ Cγ0hmaxku − u kH1(Γ) (3.150)

h 0 1 h γ max k(K + )(ψ − ψhp)k2 ≤ γ max kψ − ψhpk2 (3.151) 0 L2(Γ) 0 L2(Γ) pmin 2 pmin

∂ hp 2 hp 2 γ0hmaxk (V (ψ − ψ ))k ≤ γ0hmaxkV (ψ − ψ )k 1 ∂s L2(Γ) H (Γ)

≤ γ h kψ − ψhpk2 (3.152) 0 max L2(Γ)

75 3 Stabilized mixed hp-BEM in Linear Elasticity

1 hp 2 hmax hp 2 kγ 2 σ (u − u )k ≤ γ kσ (u − u )k n L2(ΓC ) 0 2 n L2(ΓC ) pmin h ≤ γ max kσ (u − vhp)k2 +kσ (vhp − uhp)k2 0 2 n L2(ΓC ) n L2(ΓC ) pmin (3.153)

The continuity condition of the Dirichlet-to-Neumann operator , Young’s inequality and hp hp hp we choose v = u + Πhp(u − u ) we obtain h h γ max kσ (vhp − uhp)k2 = γ max kσ (Π (u − uh))k2 0 2 n L2(Γ) 0 2 n hp L2(Γ) pmin pmin hmax h 2 ≤ γ0 2 CkΠhp(u − u )kH1(Γ) pmin h 2 ≤ Cγ0kΠhp(u − u )k 1 H 2 (Γ) h 2 ≤ Cγ0ku − u k 1 (3.154) H 2 (Γ) We employ the continuity condition of the Dirichlet-to-Neumann operator and choosing hp v = Ihpu we obtain

hmax hp 2 hmax hp 2 γ kσ (u − v )k ≤ Cγ ku − v k 1 0 2 n L2(ΓC ) 0 2 H (Γ) pmin pmin hmax 2 ≤ Cγ0 2 ku − IhpukH1(Γ) pmin 2 hmax 2 ≤ Cγ0 kuk 3 (3.155) 3 ˜ 2 pmin H (Σ) we obtain 2 hmax hp 2 hp 2 hmax 2 γ0 kσn(u − u )k = Cγ0 ku − u k 1 + kuk 3 (3.156) 2 L2(ΓC ) 2 3 ˜ 2 pmin H (Γ) pmin H (Σ) Similarly to (3.157), we get

2 hmax hp 2 hp 2 hmax 2 γ0 kσt(u − u )k = Cγ0 ku − u k 1 + kuk 3 (3.157) 2 L2(ΓC ) 2 3 ˜ 2 pmin H (Γ) pmin H (Σ)

hmax hp 2 hmax 2 hmax hp 2 γ0 ku − u kH1(Γ) ≤ γ0 ku − IhpukH1(Γ) +γ0 kIhpu − u kH1(Γ) pmin pmin pmin (3.158)

We apply the inverse inequality for the second term we obtain

2 hmax hp 2 hmax pmax hp 2 γ0 kIhpu − u kH1(Γ) ≤ γ0 kIhpu − u k 1 pmin pmin hmin H 2 (Γ) hp 2 2 ≤ Cγ0pmax ku − u k 1 +kIhpu − uk 1 (3.159) H 2 (Σ) H 2 (Σ)

76 3.4 Reliable and eﬃcient a posteriori error estimates for stabilized hp-BEM for frictional contact problems we have 2 hmax 2 hmax 2 γ0 ku − Ihpuk 1 ≤ γ0 kuk 3 (3.160) H (Γ) 2 ˜ 2 pmin pmin H (Σ) and 2 2 hmax 2 γ0pmaxku − Ihpuk 1 ≤ γ0 kuk 3 (3.161) H 2 (Γ) pmin H˜ 2 (Σ) there holds 2 hmax hp 2 hp 2 hmax 2 γ0 ku − u kH1(Γ)≤ Cγ0 pmaxku − u k 1 + kuk 3 (3.162) pmin H 2 (Σ) pmin H˜ 2 (Σ) and we have

hp 2 2 hp 2 2 2 γ0hmaxku − u kH1(Γ)≤ γ0pmaxku − u k 1 +γ0hmaxkuk 3 (3.163) H 2 (Σ) H˜ 2 (Σ)

h kψ − ψhpk2 ≤ h kψ − P ψk2 +h kP ψ − ψhpk2 (3.164) max L2(Γ) max h L2(Γ) max h L2(Γ) using the inverse inequality and lemma(3.10) for the second term

hp 2 2 hp 2 2 2 hmaxkPhψ − ψ kL (Γ) ≤ Cpmaxkψ − ψ k − 1 +CpmaxkPhψ − ψk − 1 2 H 2 (Γ) H 2 (Γ) 2 hp 2 2 2 ≤ Cpmaxkψ − ψ k 1 +Chmaxkψk 1 (3.165) H− 2 (Γ) H 2 (Γ) Employing lemma(3.10) for the ﬁrst term

2 2 hmax 2 h kψ − P ψk ≤ C kψk 1 (3.166) max h L2(Γ) pmin H 2 (Γ) we obtain hp 2 2 hp 2 2 2 γ0hmaxkψ − ψ kL (Γ)≤ Cγ0 pmaxkψ − ψ k − 1 +hmaxkψk 1 (3.167) 2 H 2 (Γ) H 2 (Γ) and 2 hmax hp 2 hp 2 hmax 2 γ kψ − ψ k ≤ Cγ p kψ − ψ k 1 + kψk 1 (3.168) 0 L2(Γ) 0 max − pmin H 2 (Γ) pmin H 2 (Γ) Finally we get X 1 2 2 hp 2 hp 2 2 Hq 2 c(γ0) ηh(I) ≤ Cpmax ku − u k 1 +kψ − ψ k − 1 +kγ (λ − λ )kL (Γ ) H 2 (Σ) H 2 (Γ) 2 C I∈Thp 2 2 2 + hmax kuk 3 +kψk 1 (3.169) H˜ 2 (Σ) H 2 (Γ)

Since un ≤ 0, we have

hp + hp 0 ≤ (un ) ≤|un − un| on ΓC , (3.170)

77 3 Stabilized mixed hp-BEM in Linear Elasticity and

hp + hp 0 ≤ k(un ) kL2(ΓC ∩I)≤ ku − u kL2(ΓC ∩I). (3.171)

Hq + Hq If I ∈ Thp, let J ⊂ I be the part of the edge where (λn ) = λn

Z Z Hq + hp + Hq hp + (λn ) (un ) ds = λn (un ) ds ΓC ∩I ΓC ∩J

Z Z Hq hp + hp + ≤ |λn − λn||(un ) | ds + |λn||(un ) | ds ΓC ∩J ΓC ∩J

Hq hp ≤ kλ − λ kL2(I∩ΓC )ku − u kL2(I∩ΓC )

hp + kλkL2(I∩ΓC )ku − u kL2(I∩ΓC ) (3.172)

Hq − Hq Noting that λn ≥ 0 on ΓC then we have 0 ≤ (λn ) ≤|λn − λn | on ΓC and we obtain

Hq − 2 Hq − 2 Hq 2 Hq k(λ ) k 1 ≤ k(λ ) k ≤ kλ − λ k ≤ kλ − λ k (3.173) n − n L2(ΓC ) n n L2(ΓC ) L2(ΓC ) H˜ 2 (ΓC )

We now estimate the term

Z Hq − hp Hq hp − |(|λt | − F) ||ut | + 2(λt ut ) ds (3.174) ΓC

Since λtut ≥ 0 on ΓC we have

Hq hp − Hq hp 0 ≤ (λt ut ) ≤|λtut − λt ut | hp Hq hp Hq ≤|λt(ut − ut ) + (λt − λt )(ut − ut ) + (λt − λt )ut| (3.175) we obtain

Z Hq hp − Hq hp (λt ut ) ds ≤ kλ − λ kL2(I∩ΓC )ku − u kL2(I∩ΓC ) I∩ΓC Hq hp + kλ − λ kL2(I∩ΓC )kukL2(I∩ΓC )+kλkL2(I∩ΓC )ku − u kL2(I∩ΓC ). (3.176)

We estimate the second term in (3.174), If I ∈ Thp, let J ⊂ I be the part of the edge where

Hq − Hq |(|λt | − F) | = (|λt | − F)

78 3.5 Numerical Experiments

Since |λt| ≤ F we obtain Z Z Hq − hp Hq hp |(|λt | − F) ||ut | ds = (|λt | − F)|ut | ds ΓC ∩I ΓC ∩J Z Hq hp = (|λt | − |λt| − F + |λt|)|ut | ds ΓC ∩J Z Hq hp ≤ (|λt | − |λt|)(|ut | − |ut|) ds+ ΓC ∩J Z Hq + (|λt | − |λt|)|ut| ds ΓC ∩J Hq hp ≤ kλ − λ kL2(I∩ΓC )ku − u kL2(I∩ΓC )

Hq + kλ − λ kL2(I∩ΓC )kukL2(I∩ΓC ) (3.177)

We now estimate the term

Hq + 2 k(|λt | − F) k − 1 (3.178) H˜ 2 (ΓC ) we have

Hq + 2 Hq + 2 k(|λ | − F) k 1 ≤ k(|λ | − F) k (3.179) t − t L2(ΓC ) H˜ 2 (ΓC )

Since |λt| ≤ F we get

Hq + Hq Hq (|λt | − F) ≤||λt | − |λt| − F + F| ≤|λt − λt|. (3.180)

Finaly we obtain

Hq + 2 Hq + 2 Hq k(|λ | − F) k 1 ≤ k(|λ | − F) k ≤ k(λ − λ )k (3.181) t − t L2(ΓC ) L2(ΓC ) H˜ 2 (ΓC ) Using the inverse inequality, we obtain

2 hp + 2 pI hp + 2 k(u ) k 1 ≤ k(u ) k . (3.182) n n L2(ΓC ∩I) H 2 (ΓC ∩I) hI From (3.171), we have

2 hp + 2 pI hp k(un ) k 1 ≤ ku − u kL2(ΓC ∩I). (3.183) H 2 (ΓC ∩I) hI

3.5 Numerical Experiments

Numerical results are presented with the MATLAB package of L.Banz for the contact 2 of the two-dimentional elastic body Ω = [−0.5, 0.5] with a rigid obstacle. ΓC =

79 3 Stabilized mixed hp-BEM in Linear Elasticity

[−0.5, 0.5] × {−0.5} and ΓN = ∂Ω \ (ΓC ∪ ΓD). The Young’s modulus and the Poisson ratio are E = 1000, ν = 0.3 respectively. The Neumann force is t = (0, 0)T , the gap |x| 5 − h g = 10 − 100 ,the stabilization parameter γ = 10 3 p2 and the give friction coeﬃcient F = 0.3. In Figure3.1 we show the initial and the deformed conﬁguration. Figure 3.2 shows the estimated errors for the h-uniform for the stabilized and the non-stabilized problems, and hp-adaptive methods for the non-stabilized problem.

80 3.5 Numerical Experiments

ΓD

ΓN ΓN

ΓC

Figure 3.1: Initial(left) and Deformed (right) geometry

2 10 uni. h, p=1, biortho hp−adaptive, biortho uni. h, p=1, stabilized

1 10

Residual Error Estimate 10

−1

10 0 1 2 3 4 10 10 10 10 10 Degrees of Freedom

Figure 3.2: Convergence for stabilized and non-stabilized problems

4 hp-BEM for Stochastic Contact Problems in Linear Elasticity

Stochastic methods are, as their name suggests, based on the use of probability to model uncertainty. The objective is to study the eﬀects on the output uncertainty on the input parameters, considered given, mechanical models. The knowledge about the uncertainties of the input data can be modeled as:

Random variables, i.e functions depending only on the hazard, so deterministic in space and time.

Stochastic ﬁelds, i.e functions depending on both the hazard and space.

Stochastic process, i.e functions depending on the hazard and time.

In this chapter we present a stochastic mixed BEM formulation for contact problems with Tresca friction.For the theoretical treatment of random variational inequalities see [29], [31] We show that the stochastic mixed formulation is well-posed. We study the deformation of an elastic homogeneous material in which Young’s modulus (parameter that characterize the material properties) is a random variable. Similary the surface force t = t(x, ω) and the gap function g are assumed to be random as well.

4.1 Mixed Formulation for Stochastic Contact Problem

2 Let D ⊂ R be a bounded Lipschitz domain with boundary Γ := ∂Ω = ΓN ∪ ΓD ∪ ΓC decomposed into the non-intersecting Neumann segment ΓN , the Dirichlet segment ΓD and the contact segment ΓC which potentially can come in contact with the rigid foundation. Further let (Ω, B,P ) be a probability space with the set of outcomes, B ⊂ 2Ω the algebra of events and P : B −→ [0, 1] a probability measure.

Let σ(u), (u) and C denote the stress tensor, the linearized strain tensor and the elasticity tensor, respectively. Further on u and n will denote the displacement ﬁeld and the outward unit normal.

We deﬁne κ(x) for each x ∈ D as a random variable κ : D × Ω −→ R on the probability space (Ω, B,P ). As a consequence κ : D × Ω −→ R is a random ﬁeld and one obtains the real number κ(x, ω) for each realization ω ∈ Ω.

83 4 hp-BEM for Stochastic Contact Problems in Linear Elasticity

Let the outcome of the elasticity tensor be a random ﬁeld C(x, .), x ∈ D, deﬁned on a probability space (Ω, B,P ).

The problem then consists in ﬁnding the displacement ﬁelds u : D × Ω −→ R such that P-a.e in Ω

− div σ(u(x, ω)) = f(x, ω) in D (4.1a) σ(u(x, ω)) = C(x, ω): (u(x, ω)) in D (4.1b)

u(x, ω) = 0 on ΓD (4.1c)

σ(u(x, ω)) · n = t(ω, x) on ΓN (4.1d)

σn ≤ 0, un(x, ω) ≤ g(x, ω), σn(un(x, ω) − g(x, ω)) = 0 in ΓC (4.1e)

|σt| ≤ F, σtut(x, ω) + F|ut(x, ω)| = 0 on ΓC (4.1f)

Bochner spaces are a generalization of the concept of Lp spaces to functions whose values lie in a Banach space which is not necessarily the space R or C of real or complex numbers.

Let a measure space (T, Σ, µ) with Σ a sigma algebra over T and µ a measure on Σ be p given, then for a Banach space (X, k· kX ), the Bochner space L (T ; X) is deﬁned as the space of all measurable functions u : T → X such that the norm

Z 1 p p kukLp(T ;X) = ( ku(t)kX dµ(t)) < +∞ for 1 ≤ p ≤ ∞ (4.2) T

kukL∞(T ;X) := ess supt∈T ku(t)kX < +∞ if p = +∞ (4.3) is ﬁnite.

In the case where p = 2 and X is a separable Hilbert space, we have by Fubini that:

2 ∼ 2 Lµ(T ; X) = L (T ) ⊗ X (4.4) where ⊗ denotes the tensor product between Hilbert spaces

2 Remark 4.1. In particular Lµ(T ; X) is itself again a Hilbert space

1 2 ˜ 2 Using the deﬁnition of Bochner spaces, let LP (Ω; H (Σ)) be the tensor Hilbert space of the second-order random variables deﬁned on the probability space (Ω, B,P ) with 1 ˜ 2 P values in H (Σ). Let Kω be the non empty closed convex subset of second-order random variables which satisfy the non-penetration condition

un ≤ g Pω − almost surely. (4.5)

1 P 2 ˜ 2 Kω := {u ∈ LP (Ω; H (Σ)) : un ≤ g Pω − a.s on ΓC × Ω}

84 4.1 Mixed Formulation for Stochastic Contact Problem

We introduce some stochastic terminology: The Karhunen-Lo`eve spectral decomposition consists in decomposing the random ﬁelds κ with eigenvalues λm in the form:

∞ X p κ(x, ω) = κ0(x) + λmbm(x)ξm(ω), (4.6) m=1 where κ0 is the mean of a random ﬁeld κ at the point x ∈ D deﬁned by Z κ0(x) := hκ(x, ·)i := κ(x, ω)dP (ω). (4.7) Ω

We approximate the random ﬁelds by a truncated Karhunen-Lo`eve expansion:

M X p κ(x, ω) ≈ κ0(x) + λmbm(x)ξm(ω), (4.8) m=1

M This determines a ﬁnite number M of independent random variable {ξm}m=1 with mean zero and unit variance.The number of random variables will be denoted M or the number of the random dimensions. These random variables are determined by 1 Z ξm(ω) = √ (κ(x, ω) − κ0(x))bm(x) dx (4.9) λm D

As a consequence, the stochastic variation of the random ﬁelds is now only through its dependence on the random variables ξ1, ··· , ξM . An assumption of ﬁnite dimensional noise must be made for each random input R(ω):

R(ω) = R(x, ξ1(ω), ··· ; ξM (ω)) =: R(x, ξ(ω)).

We denote the range space of each variable by Θm = ξm(Ω) and the product range space M Y Θ = Θm Θ ≡ Θ1 × Θ2 × · · · × ΘM . m=1 M Since we have assumed the {ξm}m=1 are independent and continuous, they have a joint density function ρ :Θ → R and ρ(y) = ρ1(y1)ρ2(y2) ··· ρM (yM ) where the ρm are the corresponding density functions of the ξm, ym ∈ Θm and y = (y1, ··· , yM ). . With this assumption by the Doob-Dynkin Lemma, the displacement u can be also described by a ﬁnite number of random variables

u(x, ω) = u(x, ξ1(ω), ··· , ξM (ω)). (4.10)

The goal of the numerical methods is to seek the solution u(x, ξ).

1 1 2 ˜ 2 2 ˜ 2 We can now replace LP (Ω; H (Σ)) by Lρ(Θ; H (Σ)) and have

1 ρ 2 ˜ 2 K := {u ∈ Lρ(Θ; H (Σ)) : un ≤ g (P ) − a.s on ΓC × Θ}.

85 4 hp-BEM for Stochastic Contact Problems in Linear Elasticity

The variational inequality of the second kind then consists in ﬁnding a random displacement u ∈ Kρ such that ∀v ∈ Kρ

A(u, v − u) + j(v) − j(u) ≥ L(v − u) ∀ v ∈ Kρ (4.11) with the bilinear form Z A(u, v) := hSyu, viΣ ρ dy, (4.12) Θ where 1 1 S := W + (K 0 + )V −1(K + ). y y y 2 y y 2 0 with Vy, Ky , Ky and Wy as in in Chapter2 for

1 1 (x1 − x2) ⊗ (x1 − x2) G(x1, x2) := {−β1log I + β2 2 } E(y) |x1 − x2| |x1 − x2| where (3 − 4ν)(1 + ν) 1 + ν β = , β = 1 2π(1 − ν) 2 4π(1 − ν) with Poisson’s ratio ν and Young’s modulus E.

The linear form Z L(v) := ht, vi ρ dy. (4.13) ΓN Θ The friction functional is given by Z Z j(v) = F|ut|ρ ds dy (4.14) Θ ΓC

1 2 ˜ 2 Deﬁne the energy functional J : Lρ(Θ; H (Σ)) → R as 1 J(v) := A(u, v) + j(v) − L(v). (4.15) 2 Remark 4.2. We denote by {C(x, · ), x ∈ D}, the random elasticity tensor ﬁeld. The bilinear form is continuous and elliptic, if C(x, · ) is uniformly bounded from below [2],[3],[1]

Lemma 4.1. The bilinear form A(· , · ) is a continuous elliptic bilinear form on

1 1 2 ˜ 2 2 ˜ 2 Lρ(Θ; H (Σ)) × Lρ(Θ; H (Σ)), i.e. there exist constants CA > 0 and cA > 0

1 2 ˜ 2 such that for all u, v ∈ Lρ(Θ; H (Σ))

A(u, v) ≤ CAkuk 1 kvk 1 (4.16) 2 ˜ 2 2 ˜ 2 Lρ(Θ;H (Σ)) Lρ(Θ;H (Σ)) 2 A(u, v) ≥ cAkuk 1 (4.17) 2 ˜ 2 Lρ(Θ;H (Σ))

86 4.2 Discretization for Stochastic Contact Problem

Proof. Follows from the fact that the Steklov-Poincar´eoperator S is continuous and ˜1 H 2 (Σ)-coercive and if the random elasticity tensor ﬁeld is uniformely bounded from below and above [2],[3],[1].

We can formulate the stochastic classical formulation as a saddle point problem equivalent to the stochastic variational inequality.

The stochastic admissible space for Lagrange multiplier λ is given by

n − 1 o L(F) = µ ∈ L2(Θ; H˜ 2 (Γ )) : R hµ, vi ρ dy ≤ R hF, kv ki ρ dy , ∀ v ∈ V − ρ C Θ ΓC Θ t ΓC where

− 2 1 V = {v ∈ Lρ(Θ; H 2 (ΓC )), vn 6 0} (4.18)

The mixed variational formulation of the stochastic contact problem with friction is given equivalently in the deterministic form by (cf.[8]):

1 2 ˜ 2 Find (u, λ) ∈ Lρ(Θ; H (Σ)) × L(F) such that

1 2 ˜ 2 A(u, v) + b(λ, v) = L(v) ∀ v ∈ Lρ(Θ; H (Σ)) (4.19a) Z b(µ − λ, v) hg, µ − λ i ρ dy ∀ µ ∈ L(F) (4.19b) 6 n n ΓC Θ with the functional

Z Z b(λ, u) = hu , λ i ρ dy + hu , λ i ρ dy n n ΓC t t ΓC Θ Θ

We deﬁne the strong pointwise non penetration condition on ΓC by

un(x, y) ≤ g(x, y), λn(x, y) ≥ 0, λn(x, y)(u(x, y)n − g(x, y)) = 0 (4.20) and the strong pointwise friction condition reads as

|λt(x, y)| ≤ F

|λt(x, y)| < F ⇒ ut(x, y) = 0 (4.21) 2 |λt(x, y)| = F ⇒ ∃α ∈ R : λt(x, y) = α ut(x, y)

4.2 Discretization for Stochastic Contact Problem

Let Thp denote a partition of Γ¯C ∪ Γ¯N , such that all corners of Γ¯C ∪ Γ¯N and all end points Γ¯C ∩ Γ¯N ,Γ¯D ∩ Γ¯N are nodes of Thp. For simplicity we assume meas(ΓC ) > 0 and

87 4 hp-BEM for Stochastic Contact Problems in Linear Elasticity

Γ¯C ∩ Γ¯D = ∅. Tkq is the mesh of Θ. Furthermore we deﬁne the set of Gauss-Lobatto points GI,hp on each element I ∈ Thp of corresponding polynomial degree pI and set

Ghp := ∪I∈Thp GI,hp.. Analogously, let Gkq be the aﬃnely transformed Gauss-Lobatto points in the stochastic domain, with Gkq := ∪J∈Tkq GJ,kq.

For p = (pI )I∈Thp associate each element of Thp with a polynomial degree pI ≥ 1.

Similary, for q = (qJ )J∈Tkq associate each element of Tkq with a polynomial degree qJ ≥ 0. In particular, the ﬁnite element discretization of the associated deterministic problem can be choosen completely independently of the stochastic discretization, The space Vhp can be spanned by the 2-d nodal basis {φiek, i = 1, ..., NV, k = 1, 2}, where ek denotes the i-th unit vector, φi the scalar Gauss-Lobatto Lagrange basis function associated with the node i and NV the total number of the nodes.

We introduce (cf.[8]) the continuous piecewise polynomial space for the discretization of u,

1 hp ˜ 2 hp 2 hp Vhp := {u ∈ H (Σ) : ∀I ∈ Thp, u |I ∈ [PpI (I)] , u = 0 on ΓD}, with dimVhp Vhp ⊂ span{φi}i=1 and the piecewise polynomial space of discrete tractions ,

T 1 dimVhp T − 2 2 T Vhp := {φ ∈ H (Σ) : ∀I ∈ Th, φ|I ∈ [PpI −1(I)] } ⊂ span{φi }i=1 .

We next introduce a ﬁnite dimensional subspace

2 dimWkq Wkq := v ∈ Ly(Θ) : vkq|J ∈ PpJ (J) ∀ J ∈ Tkq ⊂ span{ζi}i=1

1 2 ˜ 2 of the stochastic parameter space and approximate the tensor product space Lρ(Θ; H (Σ)) by the tensor product

Vhp ⊗ Wkq ⊂ span{φiζj : φi ∈ Vhp, ζj ∈ Wkq}

We deﬁne the discrete set of admissible displacements as

n hp o Khp,kq := v ∈ Vhp ⊗ Wkq :(v )ij· ni ≤ gij ,

where gij is a suitable approximation of g. We note in general that Khp,kq is not a subset of K. The weighted gap vector is given by 1 Z Z gij := stoch g(x, y)ψi(x)ζj(y) dx ρ dy, (4.22) DiDj Θ ΓC where Z Z stoch Di = φi dx, Dj = ρϑj dy. ΓC Θ

88 4.2 Discretization for Stochastic Contact Problem

Furthermore we deﬁne the dual Lagrange space

0 dim Vhp|ΓC Mhp = (Vhp|ΓC ) = span {ψj}j=1 and the stochastic dual Lagrange space

dim Wkq Tkq := span {ζj}j=1 . The discrete Lagrange multiplier space is given by Z Z hp hp hp hp hp − Lhp,kq := µ ∈ Mhp ⊗ Tkq : hµ , v iρ dy ≤ hF, |vt |hiρ dy, ∀v ∈ Vhp,kq , Θ Θ (4.23) where

− Vhp,kq := {v ∈ Vhp ⊗ Wkq, vn,ij ≤ 0}.

The dual or biorthogonal basis functions ψi, ζj satisfy the orthogonality relations Z Z Z Z ψiφjds = δij φjds, ζiϑjρ dy = δij ϑjρ dy. (4.24) ΓC ΓC Θ Θ

hp The discrete function u ∈ Vhp ⊗ Wkq then is of the form ( for 1 ≤ i ≤ dimVhp and 1 ≤ j ≤ dimWhp)

hp X u := uijφi(x)ϑj(y) ij X := (un,ijni + ut,ij)φi(x)ϑj(y). (4.25) ij

hp The normal and tangential part of the discrete function u ∈ Vhp ⊗ Wkq are given by

hp X un := un,ijφi(x)ϑj(y), (4.26) ij hp X ut := ut,ijφi(x)ϑj(y). (4.27) ij

The discrete absolute value on ΓC of the tangential component is

hp X |ut |h := |ut,ij|φi(x)ϑj(y). ij

Similary, the discrete Lagrange multiplier is given by (for 1 ≤ i ≤ dimMhp and 1 ≤ j ≤ dimThp)

hp X λ := λijψi(x)ξj(y) (4.28) ij X := (λn,ijni + λt,ij)ψi(x)ξj(y). (4.29) ij

89 4 hp-BEM for Stochastic Contact Problems in Linear Elasticity

We can express the normal and the tangential part of λhp as follows:

hp X λn := λn,ijψi(x)ξj(y), (4.30) ij hp X λt := λt,ijψi(x)ξj(y). (4.31) ij

Lemma 4.2. The space Lhp,kq (4.23) is equivalent to    hp X  Lhp,kq := µ := µijψiξj, µn,ij ≥ 0, |µt,ij| ≤ F, 1 ≤ i ≤ dim Mhp, 1 ≤ j ≤ dim Thp  ij  (4.32)

Proof. ” ⇒ ” By means of the biorthogonality relations Z Z φiψjds = δijDi,Di = φids > 0, (4.33) ΓC ΓC Z Z stoch stoch ϑiξj ρ(y)dy = δijDi ,Di = ϑi ρ(y)dy > 0. (4.34) Θ Θ hp hp µ ∈ Lhp,kq and v ∈ Vhp ⊗ Wkq can be written as

hp X v := (vn,ijni + vt,ij)φi(x)ϑj(y), (4.35) ij hp X µ := (µn,ijni + µt,ij)ψi(x)ξj(y). (4.36) ij Similary hp X µn := µn,ijψi(x)ξj(y), ij and the norm of the tangential component of vhp is

hp X |vt |h := |vt,ij|φi(x)ϑj(y) ij

µn,ijvn,ij + µt,ijvt,ij ≤ F|vt,ij|. (4.37)

90 4.2 Discretization for Stochastic Contact Problem

hp Since vn ≤ 0, we get vn,ij ≤ 0. Choosing vt,ij = 0 in (4.37)

µn,ijvn,ij ≤ 0, and we get

µn,ij ≥ 0

Choosing now vt,ij = µt,ij and vn,ij = 0 in (4.37) , we obtain

|µt,ij| ≤ F.

” ⇐ ” We have Z hp hp X stoch hµ , v iρ dy = (µn,ijvn,ij + µt,ijvt,ij)DiDj (4.38) Θ ij and Z hp X stoch hF, |vt |hiρ dy = F |vt,ij|DiDj . (4.39) Θ ij

hp Since vn,ij ≤ 0, we get for each µ ∈ Lhp,kq

µn,ijvn,ij + µt,ijvt,ij ≤ |µt,ij||vt,ij|

≤ F|vt,ij|.

Summing over all points, we obtain the assertion, (see[[41],Lemma2.3]).

The approximation Shp of the Poincar´e-Steklov operator is given by 1 1 S := W + (K0 + )i (i∗ V i )−1i∗ (K + ). (4.40) hp 2 hp hp hp hp 2

hp The discrete variational inequality reads: Find u ∈ Khp,kq such that

hp hp hp hp hp hp hp Ah(u , v ) + j(v ) − j(u ) ≥ L(v − u ) ∀v ∈ Khp,kq, (4.41a) where Z Z hp hp j(v ) = F|ut (x, y)|ρ ds dy. Θ ΓC

The discrete bilinear form is Z Z hp hp hp hp hp 0 1 hp Ah(u , v ) := hShpu , v iΣ ρ dy = W u + (K + )Ψhp, v ρ dy, Θ Θ 2 Σ (4.42)

T and Ψhp ∈ Vhp solves for P-a.e. y ∈ Θ the auxiliary problem D hpE 1 hp hp hp T V Ψhp, v = (K + )u , v ∀ v ∈ Vhp. (4.43) Σ 2 Σ

91 4 hp-BEM for Stochastic Contact Problems in Linear Elasticity

1 hp hp 2 ˜ 2 Lemma 4.3. The discrete bilinear form Ah(u , v ) is continuous and Lρ(Θ; H (Σ))- coercive.

The discrete mixed stochastic problem in deterministic formulation reads : Find uhp ∈ hp Vhp ⊗ Wkq and λ ∈ Lhp,kq

hp hp hp hp hp hp Ah(u , v ) + b(λ , v ) = L(v ) ∀ v ∈ Vhp ⊗ Wkq (4.44a) Z hp hp hp D hp hpE hp b(µ − λ , u ) 6 g, µn − λn ρ dy ∀ µ ∈ Lhp,kq (4.44b) Θ ΓC with the functional Z Z hp hp D hp hpE D hp hpE b(u , λ ) = un , λn ρ dy + ut , λt ρ dy. Θ ΓC Θ ΓC Theorem 4.1. There exists exactly one solution to the discrete mixed formulation (4.44).

hp hp hp hp Proof. Uniquiness: We assume that (u1 , λ1 ) and (u2 , λ2 ) solve the discrete mixed formulation (4.44). Then we have

hp hp hp hp hp hp hp hp Ah(u1 − u2 ), u1 − u2 ) + b(λ1 − λ2 , u1 − u2 ) = 0, (4.45) where Z hp hp hp hp Ah(u , v ) := hShpu , v iΣ ρ dy Θ Z Z hp hp D hp hpE D hp hpE b(u , λ ) = un , λn ρ dy + ut , λt ρ dy. (4.46) Θ ΓC Θ ΓC

hp hp hp hp Choosing µ1 = λ2 and µ2 = λ1 in (4.44b) we get

hp hp hp hp b(λ1 − λ2 , u1 − u2 ) ≥ 0. (4.47)

Using Lemma4.3 we obtain

hp hp 2 hp hp hp hp hp hp hp hp cku − u k 1 ≤ Ah(u − u ), u − u ) + b(λ − λ , u − u ) = 0 1 2 2 ˜ 2 1 2 1 2 1 2 1 2 Lρ(Θ;H (Σ)) (4.48)

Consequently the ﬁrst argument uhp is unique.

hp hp Since u is unique we have for all v ∈ Vhp ⊗ Wkq

hp hp hp 0 = b(λ1 − λ2 , v ). (4.49)

hp hp hp Using λ1 , λ2 and v and the biorthogonality , we obtain the relation

X stoch 0 = [(λ1,n,ij − λ2,n,ij)vn,ij + (λ1,t,ij − λ2,t,ij)vt,ij]DiDj (4.50) ij

92 4.2 Discretization for Stochastic Contact Problem

Choosing vt,ij = 0 in (4.50) we get

(λ1,n,ij − λ2,n,ij)vn,ij = 0 and λ1,n,ij = λ2,n,ij

stoch because DiDj > 0.

We choose now vn,ij = 0, and obtain λ1,t,ij = λ2,t,ij. Consequently the second argument λhp is unique.

Existence: We know that the inequality (4.44b) can be written as a projection equation [44]:

hp !! hp hp un − g λ = PLhp,kq λ + r hp , (4.51) ut

where the map PLhp,kq stand for the orthogonal projection onto Lhp,kq and where r > 0 is an arbitrary parameter. The ﬁxed point operator T is deﬁned as follows [44] :

T : Lhp,kq −→ Lhp,kq hp !! hp hp un − g λ 7−→ PLhp,kq λ + r hp . (4.52) ut

From Lemma 4.3 we have

hp hp 2 hp hp 2 −ku − u k 2 ≥ −ku − u k 1 1 2 Lρ(Θ;L2(ΓC )) 1 2 2 ˜ 2 Lρ(Θ;H (Σ)) Z −1 hp hp hp hp ≥ −c hShp(u1 − u2 ), u1 − u2 iΣ ρ dy (4.53) Θ

hp hp hp hp hp hp Using the notation δu = u − u , δλ = λ − λ , k·k= k·k 2 , we 1 2 1 2 Lρ(Θ;L2(ΓC )) compute:

hp !! hp !! hp hp 2 hp un,1 − g hp un,2 − g 2 kT (λ1 ) − T (λ2 )k ≤ kPLhp,kq λ1 + r hp − PLhp,kq λ2 + r hp k ut,1 ut,2 hp ! hp δun 2 ≤ kδλ + r hp k δut ≤ kδλhp + rδuhpk2 ≤ kδλhpk2+2rb(δλhp, δuhp) + r2kδuhpk2 Z hp 2 hp hp 2 hp 2 ≤ kδλ k −2r hShpδu , δu iΣ ρ dy + r kδu k Θ ≤ kδλhpk2−2rckδuhpk2+r2kδuhpk2 ≤ kδλhpk2(1 − 2rcβ2 + r2β2)

kδλhpk2 where β = kδuhpk2 . It follows that T is strict contraction for 0 < r < 2c. By the hp hp hp Banach ﬁxed point theorem there exists a λ = (λn , λt ) which satisﬁes (4.52). For

93 4 hp-BEM for Stochastic Contact Problems in Linear Elasticity any given λhp, problem (4.44a) reduces to a linear, ﬁnite dimentional problem. Hence, the uniqueness result of uhp implies the existence of a uhp(λhp).

Remark 4.3. Theorem 4.1 is the system case corresponding to the scalar problem considered in [8]. But Theorem 4.1 needs a new proof. Lemma 4.4. The contact constraints (4.44b) are equivalent to the pointwise conditions

un,psp,pst ≤ gpsp,pst , λn,psp,pst ≥ 0, λn,psp,pst (un,psp,pst − gpsp,pst ) = 0 (4.54)

|λt,psp,pst | ≤ Fλn,ps,pst

|λt,psp,pst | < Fλn,ps,pst ⇒ un,psp,pst = 0 (4.55) 2 2 |λt,psp,pst | = Fλn,ps,pst ⇒ α ∈ R : λt,psp,pst = α un,psp,pst

∀ (psp, pst) ∈ Ghp ∩ ΓC × Gkq.

Proof. see [41], [10] and [8]

For Tresca’s friction law the friction coeﬃcient is given by a function F(· ):ΓC → R.

We deﬁne Fpsppst associated with the nodes (psp, pst) ∈ Ghp ∩ ΓC × Gkq ∩ Θ by Z Z

Fpsppst = F φpsp (x)ϑpst (y) ρ dy ds, (4.56) s Θ ΓC If F is a constant function we have

Fpsppst = F Dpsp Dpst . (4.57) We will now formulate the classical Uzawa algorithm for problem (4.44). As in [8],[44], we introduce the following equivalent formulation:

Lemma 4.5. The pair (λn,psp,pst , un,psp,pst ) satisﬁes the pointwise non-penetration condition (4.54) if and only if it satisﬁes the pointwise condition

λn,psp,pst = max{0, λn,psp,pst + c(un,psp,pst − gpsp,pst )} (4.58)

Proof. see [[41],Theorem 4.1]

Lemma 4.6. The pair (λt,psp,pst , ut,psp,pst ) satisﬁes the friction contact condition (4.55) if and only if it satisﬁes the pointwise condition

λt,psp,pst + c ut,psp,pst λt,psp,pst = Fij (4.59) max{Fps,pst , |λt,psp,pst + cut,ps,pst |}

94 4.3 A posteriori error estimates for stochastic contact with friction

Proof. see [[41],Theorem 5.1]

The Uzawa algorithm corresponds to iterations of the ﬁxed point operator. In this algorithm, we use the pointwise condition (4.58) and (4.59) . The Uzawa algorithm can be written as follows:

Algorithm 4.1. (Uzawa algorithm)

1. Initialisation: Choose initial solution λ0, c > 0, tol > 0

2. For k = 0, 1, 2, ... do

a) For given λk , ﬁnd uk by solving (4.44a) and (4.44b)

b) Update Lagrange multiplier by using

λn,psp,pst = max{0, λn,psp,pst + c(un,psp,pst − gpsp,pst )}

λt,psp,pst + c ut,psp,pst λt,psp,pst = F max{Fps,pst , |λt,psp,pst + cut,ps,pst |}

c) Stop if kλk−1 − λkk< tolkλk−1k or kλk−1 − λkk< tol, else go to step 2

4.3 A posteriori error estimates for stochastic contact with friction

1 ˜ 2 Lemma 4.7. [[19], Lemma 3.2.9] There exists an operator Πhp : H (Σ) → Vhp, which 1 2 is stable in the H˜ -norm and has quasioptimal approximation properties in the L2- norm. More precisely, there exists a constant C, independent of h and p, such that for 1 all u ∈ H˜ 2 (Σ) there holds

kΠhpuk 1 ≤ Ckuk 1 (4.60) H˜ 2 (Σ) H˜ 2 (Σ) (1−) h 2 ku − ΠhpukL (Σ) ≤ C kuk 1 (4.61) 2 p H˜ 2 (Σ)

1 with arbitrarily small ∈ (0; 2 ).

Theorem 4.2. Let (u, λ) be the exact solution of the boundary problem (4.19) and (uhp, λhp) be the solution of the discrete boundary problem (4.41), then there holds the

95 4 hp-BEM for Stochastic Contact Problems in Linear Elasticity a posteriori estimate :

hp 2 hp X X 2 ku − u k 1 + kλ − λ k − 1 ηhp(I,J) 2 ˜ 2 2 ˜ 2 . Lρ(Θ;H (Σ)) Lρ(Θ;H (ΓC )) I∈Thp J∈Tkq Z hp + hp + hp − 2 + h(λ ) , (g − u ) iΓ ρ dy + k(λ ) k 1 n n C n 2 ˜ − Θ Lρ(Θ;H 2 (ΓC )) hp + 2 hp + 2 + k(u − g) k 1 +k(|λ | − F) k 1 n 2 t 2 ˜ − Lρ(Θ;H 2 (ΓC )) Lρ(Θ;H 2 (ΓC )) Z Z hp − h hp hp − + |(|λt | − F) ||ut | + 2(λt ut ) ds ρ dy (4.62) Θ ΓC where 1−! 2 hI hp 2 hp hp 2 η = 1 + kt − S u k 2 +k(−λ ) − S u k 2 hp hp L (J;L2(I∩Γ )) hp L (J;L2(I∩Γ )) pI ρ N ρ C

∂ ∗ hp 2 + hI k (V (ψhp − ψ ))k 2 , (4.63) ∂s Lρ(J;L2(I)) 1 with arbitrarily small ∈ (0; 2 ).

Proof. Using the same arguments as in Chapter 2.

From (2.134) we have Z hp 2 hp 2 hp hp CW ku − u k 1 + CV kψ − ψ k 1 ≤ ht − Shpu , u − v iΓN ρ dy 2 ˜ 2 2 ˜ − 2 Lρ(Θ;H (Σ)) Lρ(Θ;H (Γ)) Θ Z hp hp hp + h(−λ ) − Shpu , u − v iΓC ρ dy Θ Z Z hp hp hp hp + hλn − λn , un − uni ρ dy + hλt − λt , ut − uti ρ dy Θ Θ Z ∗ hp hp + hV (ψhp − ψ ), ψ − ψ i ρ dy Θ + A + B + C (4.64)

We estimate the ﬁrst and the second terms in (4.64), employing the Cauchy-Schwarz inequality. We obtain Z Z hp hp hp hp hp A = ht − Shpu , u − v iΓN ρ dy + h(−λ ) − Shpu , u − v iΓC ρ dy ≤ Θ Θ X X hp hp hp k(−λ ) − S u k 2 ku − v k 2 hp Lρ(J;L2(I)) Lρ(J;L2(I)) J∈Tkq I∈Thp∩ΓC X X hp hp + kt − S u k 2 ku − v k 2 (4.65) hp Lρ(J;L2(I)) Lρ(J;L2(I)) J∈Tkq I∈Thp∩ΓN

2 Let πkq the Lρ-projection onto Wkq, which satisﬁes Z (πkqw − w) v ρ dy. (4.66) Θ

96 4.3 A posteriori error estimates for stochastic contact with friction

2 Note that πkq is Lρ-stable, using Lemma4.7, we obtain

2 2 2 kw − πkqΠhpwk 2 2 ≤ kw − Πhpwk 2 2 +kΠhpw − πkqΠhpwk 2 2 Lρ(Θ;L (I)) Lρ(Θ;L (I)) Lρ(Θ;L (I)) 1− hI 2 2 ≤ C kwk 1 +kΠhpwk 2 2 2 ˜ 2 Lρ(Θ;L (I)) pI Lρ(Θ;H (ω(I))) 1− hI 2 2 ≤ C kwk 1 +kwk 1 2 ˜ 2 2 ˜ 2 pI Lρ(Θ;H (ω(I))) Lρ(Θ;H (ω(I))) 1−! hI 2 ≤ C 1 + kwk 1 , (4.67) 2 ˜ 2 pI Lρ(Θ;H (ω(I)))

1 ˜ 2 where we used the H -stability of Πhp.

We have

1−! 2 hI 2 kw − πkqΠhpwk 2 2 ≤ C 1 + kwk 1 , (4.68) Lρ(Θ;L (I)) 2 ˜ 2 pI Lρ(Θ;H (ω(I)))

hp hp hp Choosing v = u + πkqΠhp(u − u ), we get

Z Z hp hp hp hp hp A = ht − Shpu , u − v iΓN ρ dy + h(−λ ) − Shpu , u − v iΓC ρ dy ≤ Θ Θ 1 1−! 2 X X hI hp hp hp 1 + k(−λ ) − Shpu k 2 ku − u k 1 Lρ(J;L2(I)) 2 ˜ 2 pI Lρ(J;H (Σ)) J∈Tkq I∈Thp∩ΓC 1 1−! 2 X X hI hp hp + 1 + kt − Shpu k 2 ku − u k 1 Lρ(J;L2(I)) 2 ˜ 2 pI Lρ(J;H (Σ)) J∈Tkq I∈Thp∩ΓN (4.69)

As in (2.138) there holds for the last term C

Z ∗ hp hp ∗ hp hp hV (ψ − ψ ), ψ − ψ i ρ dy ≤ kV (ψ − ψ )k 1 kψ − ψk 1 hp hp 2 ˜ 2 2 ˜ − 2 Θ Lρ(Θ;H (Σ)) Lρ(Θ;H (Σ))   1 Z 2 X ∂ ∗ hp 2 hp ≤ c hI k (V (ψ − ψ ))k kψ − ψk 1 (4.70)  hp L2(I) 2 ˜ − 2 Θ ∂s Lρ(Θ;H (Σ)) I∈Thp

97 4 hp-BEM for Stochastic Contact Problems in Linear Elasticity

From (2.140) and (2.141), we obtain Z Z hp hp hp hp B = hλn − λn , un − uni ρ dy + hλt − λt , ut − uti ρ dy Θ Θ hp − hp ≤ k(λ ) k 1 ku − u k 1 n 2 ˜ − n n 2 Lρ(Θ;H 2 (ΓC )) Lρ(Θ;H 2 (Σ)) hp hp + + kλ − λ k 1 k(u − g) k 1 n n 2 ˜ − n 2 Lρ(Θ;H 2 (ΓC )) Lρ(Θ;H 2 (ΓC )) Z hp + hp + + h(λn ) , (g − un ) iΓC ρ dy Θ hp + hp + k(|λ | − F) k 1 ku − u k 1 t 2 ˜ − t t 2 Lρ(Θ;H 2 (ΓC )) Lρ(Θ;H 2 (Σ)) Z Z hp − h hp hp − + |(|λt | − F) ||ut | + 2(λt ut ) ds ρ dy. (4.71) Θ ΓC Now we combine estimates (4.69), (4.70), and (4.71), use the Cauchy-Schwarz inequality, and Young’s inequality, we obtain

hp 2 hp 2 X X 2 ku − u k 1 +kψ − ψ k 1 ηhp(I,J) 2 ˜ 2 2 ˜ − 2 . Lρ(Θ;H (Σ)) Lρ(Θ;H (Γ)) I∈Thp J∈Tkq Z hp + hp + hp − 2 + h(λ ) , (g − u ) iΓ ρ dy + k(λ ) k 1 n n C n 2 ˜ − Θ Lρ(Θ;H 2 (ΓC )) hp + 2 hp + 2 + k(u − g) k 1 +k(|λ | − F) k 1 n 2 t 2 ˜ − Lρ(Θ;H 2 (ΓC )) Lρ(Θ;H 2 (ΓC )) Z Z hp − h hp hp − + |(|λt | − F) ||ut | + 2(λt ut ) ds ρ dy Θ ΓC hp + kλ − λk 1 , (4.72) 2 ˜ − 2 Lρ(Θ;H (ΓC )) where 1−! 2 hI hp 2 hp hp 2 η = 1 + kt − S u k 2 +k(−λ ) − S u k 2 hp hp L (J;L2(I∩Γ )) hp L (J;L2(I∩Γ )) pI ρ N ρ C

∂ ∗ hp 2 + hI k (V (ψhp − ψ ))k 2 (4.73) ∂s Lρ(J;L2(I)) As in Lemma2.13, using estimate (4.68), we obtain hp 0 hp 2 hp 2 kλ − λk − 1 ≤ C ku − u k 1 +kψ − ψ k 1 2 ˜ 2 2 ˜ 2 2 ˜ − 2 Lρ(Θ;H (ΓC )) Lρ(Θ;H (Σ)) Lρ(Θ;H (Γ)) 00 X X 2 + C ξhp(I,J), (4.74) I∈Thp J∈Tkq where 1−! 2 hI hp 2 hp hp 2 ξ = 1 + kt − S u k 2 +k(−λ ) − S u k 2 . hp hp L (J;L2(I∩Γ )) hp L (J;L2(I∩Γ )) pI ρ N ρ C (4.75)

Combining (4.72) and (4.74) yields the aposteriori error estimate.

98 4.4 Numerical Experiments

4.4 Numerical Experiments

In this section, we consider a random position-independent Young’s modulus E and a deterministic position-independent Poisson ratio ν = 0.3. This enables the tensor product of matrices, the global matrix can be written as tensor product of the matrix representation of the Steklov-Poicar´eoperator S˜ and the stochastic mass matrix. Where S˜ is the Steklov-Poincr´eoperator for E = 1 and 1 1 S := W + (K0 + )V −1(K + ) = ES˜ 2 2

Numerical results are presented for the contact of the two-dimensional elastic body D = 2 [−0.5, 0.5] , with a rigid obstacle.The rigid obstacle occupies the half space x2 < −0.5, Γ = [−0.5, 0.5] × {−0.5} . We use the uniform distribution, which corresponds to the C √ √ density ρ ≡ √1 and the stochastic domain Θ = − 3, 3. The Neumann force is 2 3 t = (0, −1)T , the gap g = −0.2 and the give friction coeﬃcient F = 0.3. Mean values of the Young modulus E was adopted as 2000P a. The random Young’s modulus is modeled by a unifom random variable with values in the interval between 1800 and 2200.

1 In Figure 4.2 we present the error in the energy J(u) := 2 hu,Sui − ht, ui and in the energy norm J˜(u) := hu,Sui with respect to the number of the degrees of freedom, for h- uniform and p-version. The exact value of the potential J(u) ≈ 28.6199, J˜(u) ≈ 57.2398 are obtained from extrapolation of the potential values of the lowest order h-version for p = 1 and q = 0. In Figure4.1 we show the deformed conﬁguration for the mean value of Young modulus E. To solve the nonlinear equation, we require more than 300 Uzawa iterations to achieve a tolerance of 10−10.

99 4 hp-BEM for Stochastic Contact Problems in Linear Elasticity

Figure 4.1: Deformed geometry with the mean value of Young modulus E

Figure 4.2: Convergence of Stochastic Contact problem

100 5 Extended Multiscale Finite Element Method in Linear Elasticity

In this chapter, an extended multiscale finite element method EMsFEM is derived for the analysis of linear elastic heterogenous materials. The main idea is to construct numerically a finite element basis functions that captures the small-scale information ( the fine mesh ) within each coarse element [26], [54]. The construction of the basis functions is done separately for each coarse element with a linear boundary condition. The boundary conditions for the construction of the multiscale basis functions have a big influence on capturing the smale-scale information. We analyse a corresponding FEM/BEM coupling and derive an a priori error and a-posteriori error estimate. Next we present finite element implementations for nonperiodic case.

5.1 The equation of linear elasticity

In this section we consider a two-dimensional plan strain deterministic problem

2 Let Ω be a bounded domain in R with polygonal boundary ∂Ω in which every point T is represented by cartesian coordinate x = (x1, x2) . We consider a solid body in Ω deformed under the inﬂuence of a volume force f and a tension force t.

The displacement ﬁeld u of the body is governed by the linear elasticity system:

− div σ(u) = f in Ω (5.1a) σ(u) = C : ε(u) in Ω, (5.1b) where σ is the stress tensor, the strain tensor ε is given by the symmetric part of the deformation gradient 1 ε(u) = (∇u + ∇uT ) (5.2) 2 C = C(x), x ∈ Ω is the 4-th order elasticity tensor, it describes the elastic stiﬀness of the material under load.

The system given in equation (5.1) follows the boundary conditions

u = g on ΓD (5.3a)

σ(u) · n = t on ΓN , (5.3b)

101 5 Extended Multiscale Finite Element Method in Linear Elasticity where n is the unit outer normal vector on ∂Ω.

The boundary Γ = ∂Ω be decomposed into two disjoint subsets ΓD and ΓN , such that Γ = ΓD ∪ ΓN and meas(ΓD) > 0.

2 Definition 5.1. Define the Sobolev space for vector fields in R by V = H1(Ω) = [H1(Ω)]2

1 1 where u(x) ∈ H (Ω) means that ui(x) ∈ H (Ω) for any i = 1, 2, and H1(Ω) = {f ∈ L2(D): ∂2f ∈ L2(Ω)} equiped with the norm   1 Z 2 X 2 kukH1(Ω)=  |∂ u|dx s≤1 Ω

2 where s = (s1, s2) ∈ N is a multiindex with |s| = s1 + s2. We deﬁne the following continuous functions spaces

1 V0 = {u ∈ H (Ω); u = (u1, u2): u = 0 on ΓD} ⊂ V we deﬁne the bilinear form Z a(u, v) := ε(u): C : ε(v) dx. (5.4) Ω This form is symmetric, continuous and coercive, the coercivity i.e

∃ C0 > 0 : a(u, v) ≥ C0kukH1(Ω) ∀u ∈ V The variational formulation is given by

a(u, v) = F (v) ∀v ∈ V0, (5.5) where

Z Z F (v) = f v dx + t v ds (5.6) Ω ΓN

5.2 The ﬁnite element discretization

2 Let Th be a quasi-uniform triangulation of Ω ⊂ R , with the mesh parameter h and let Nh be the set of vertices of Th contained in Ω, we denote the number of grid points in Nh by np.

102 5.2 The ﬁnite element discretization

np Let {ϕj}j=1 be the system of piecewise linear basis functions on the triangulation Th of Ω such that ϕi(xj) = δij ∀xj ∈ Nh.

h Space V0 is then replaced by their discrete approximation V0 , the discrete solution uh is given by: h uh = (u1h, u2h) ∈ V0 where

np np X j X j α uαh = uα ϕj and uh = uα ϕj e α = 1, 2 (5.7) j=1 j=1 Then we deﬁne the basis function

α 2 φj(α) = ϕj e :Ω −→ R (5.8) of Vh as a vector ﬁeld with a scalar nodal function in one of their components and zero in the others.

We set 1 1 2 2 np np U = (u1, u2, u1, u2, . . . , u1 , u2 ) j j where uα are nodal values of uh i.e uαh(xj) = uα.

In the two-dimentional problem nd = 2np denotes the total number of degrees of freedom h of V0 .

h The discretization form is : Find uh ∈ V such that h a(uh, vh) = F (vh) ∀vh ∈ V (5.9) where

Z a(uh, vh) = ε(uh): C : ε(vh) dx (5.10) Ω and Z Z F (vh) = f vh dx + t vh ds (5.11) Ω ΓN Integrals in (5.9) are computed as sums of integrals over all element T using the fact that Ω = ∪T ∈Th T X Z X Z X ε(uh): C : ε(vh) dx = f vh dx t vh ds T T T ∈Th T ∈Th E∈ΓN We write

 ∂u1  ∂x1 ∇u =  ∂u2   ∂x2  ∂u1 + ∂u2 ∂x1 ∂x2

103 5 Extended Multiscale Finite Element Method in Linear Elasticity

The equation (5.9) can be written as     ∂u1 ∂v1 ! ! Z ∂x1 ∂x1 Z v Z v  ∂u2   ∂v2  1 1 ∂x C ∂x = (f1, f2) + (t1, t2)  2   2  v2 v2 Ω ∂u1 + ∂u2 ∂v1 + ∂v2 Ω ΓN ∂x1 ∂x2 ∂x1 ∂x2 The corresponding linear algebra equation serie is given by AU = b.

We introduice the matrix Bi related to a node xi by

  ∂ϕi 0 ∂x1 ∂ϕi Bi =  0   ∂x2  ∂ϕi ∂ϕi ∂x1 ∂x2

The bilinear form in equation (5.9) applied to the basis function of Vh reads Z α > β a(φi(α), φj(β)) = ε(ϕie ) Cε(ϕje ) dx (5.12) Ω 2 Z X = Cαrβl∂lϕj∂rϕi dx (5.13) Ω r,l=1

nd=2np nd×nd and we deﬁne the stiﬀness matrix A = (aij)i,j=1 ∈ R It holds α α ε(φi(α)) = ε(ϕie ) = Bie where α φi(α) = Bie We can write Z α > > β Ai(α)j(β) = (e ) Bi C Bj dx e α, β = 1, 2 (5.14) Ω

For each element T ∈ Th we deﬁne the stiﬀness matrix Z > AT = BT C BT dx (5.15) T where the matrix BT contains the nodal matrices BTi , i = 1, 2, 3, 4 corresponding to the 4 vertices of T

BT = [BT1 ,BT2 ,BT3 ,BT4 ] (5.16)

Elementary calculations provide on element T   ∂ϕ1 0 ∂ϕ2 0 ∂ϕ3 0 ∂ϕ4 0 ∂x1 ∂x1 ∂x1 ∂x1 ∂ϕ1 ∂ϕ2 ∂ϕ3 ∂ϕ4 3×8 BT =  0 0 0 0  ∈  ∂x2 ∂x2 ∂x2 ∂x2  R ∂ϕ1 ∂ϕ1 ∂ϕ2 ∂ϕ2 ∂ϕ3 ∂ϕ3 ∂ϕ4 ∂ϕ4 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1

104 5.3 Extended multiscale ﬁnite element method for the analysis of linear elastic heterogeneous materials

5.3 Extended multiscale ﬁnite element method for the analysis of linear elastic heterogeneous materials

We present an approach where, instead of approximating in VH , we use a better space Ms 1 of multiscale functions VH ⊂ H0(Ω). The multiscale ﬁnite element solution found Ms from solving the ﬁnite element problem using VH .

Ms Ms The space VH is the span of the set of multiscale basis fonction {Φi } which are defined for each node of a coarse mesh TH (Ω) .The idea of the method is to construct numerically the multiscale basis functions to capture the fine scale features of the coarse elements in the multiscale finite element analysis.

In this section, we also give the deﬁnitions of the multiscale basis and the multiscale coarse space .

2 We consider a two dimentional plan stain problem, let Ω be a bounded domaine in R with polygonal boundary ∂Ω = Γ where ΓD and ΓN are the segments on which we T prescribe homogeneous Dirichlet and Neumann boundary conditions and u = [ux, uy] ist the displacement ﬁeld, f and t are the body forces and tractions respectively and C is the symmetric elasticity tensor and n is the outward normal on ∂Ω, it is assumed 2 2 that f ∈ L (Ω) and t ∈ L (ΓN ).

The weak formulation then consists in ﬁnding u ∈ V, where

1 1 2 V := {v ∈ H (Ω) = [H (Ω)] : v|ΓD = 0} such that

a(u, v) = F (v) ∀ v ∈ V (5.17) where a : V × V −→ R and F : V −→ R are bilinear and linear form on V deﬁned as Z Z Z a(u, v) := σ(u): ε(v) dx and F (v) := f · v dx + t · v ds (5.18) Ω Ω ΓN The solution u ∈ V is called the weak solution to the boundary value problem and Lax-Milgran lemma ensures its existence and uniqueness.

Let TH be the coarse triangulation of Ω, here we assume again that each coarse element T consists of union of fine elements τ ∈ Th of the fine triangulation. The coarse elements T ∈ TH are constructed by agglomeration of the fine elements, we construct a set of agglomerated elements {T } = TH such that each T = ∪τ∈Th τ is a simply connected union of fine grid elements, let ΣH the coarse grid points in Ω.

p For each coarse node x ∈ ΣH we denote the k-th coarse degree of freedom for k ∈ p {1, 2} related to this node by p(k), we denote wp = {T ∈ TH : x ∈ T } the union p of quadrilaterals connected to the node x and Hwp = diam(wp), we enummerate its four vertices by p = 1, 2, 3, 4 and for every node xp on the Dirichlet boundary, we

105 5 Extended Multiscale Finite Element Method in Linear Elasticity

also denote Γp,D = wP ∩ ΓD and for every element T in TH we denote by wT the 0 0 patch of quadrilaterals containing T that is wT = {∪T : T ∪ T 6= ∅} ∪ T and set 1 1 1 2 2 2 HwT = diam(wT ), it holds that |T | ∼ HT , |wp| ∼ Hwp and |wT | ∼ HwT .

The shape regularity of the mesh TH imposed, HE = |E| ∼ HT for every edge E of T, whereas the local quasi-uniformity implies in particularity that HT ∼ Hwp ∼ HwT .

5.3.1 The construction of the basis functions

Ms 2 Ms We construct a vector valued multiscale basis function Φp(k) : wp −→ R where Φp(k) is the basis associed with the node p and supported on wp. The construction is done Ms separatly for each element T ∈ TH ,Φp(k) is used for the displacement in the direction k.

p lin We denote the scalar coarse nodal basis function corresponding to x by φp : wp −→ R, lin lin q q Ms φp is linear in T and φp (x ) = δpq, with x ∈ ΣH the basis functions Φp(k) whose Ms restriction Φp(k),T to T must solve the following subgrid problem:

Ms 1 Find Φp(k),T ∈ H (T ) such that

Ms 1 aT (Φp(k), v) = 0 for all v ∈ H (T ),T ⊂ wp, (5.19) subject to a suitable boundary condition

Ms lin k Φp(k),T = φp,T · e on ∂T ; T ⊂ wp, (5.20) where

X Ms k Φp(k) = 1Ωe on ∂T. (5.21) p x ∈ΣH (T ) On ∂T , linear boundary conditions are imposed in the k-th component of the vector-ﬁeld and zero boundary conditions in the {1, 2}\{k}.

The local boundary conditions will be constructed so that they are continuous across element edges, that is

Ms Ms lin k Φp(k),T (x) = Φp(k),T 0 (x) = φp,T · e for x ∈ ∂T ∩ ∂T 0 . For the two-dimensional problem two kinds of basis functions are constructed one is used for the x-axis direction and the other is used for the y-axis x direction. Firstly, let us consider the construction of the basis function Φp1 (x) = Φp1x on node 1 of the coarse element Figure 5.1 (left). The displacement at all boundary nodes are not constraint in y-axis direction except node 3, for which the displacement are ﬁxed to zero in both coordinate directions in order to avoid rigid displacement, a unit displacement is applied on node 1 in the positive x-direction Figure 5.1 (left). The displacement at nodes 2, 3 and 4 are ﬁxed to zero in x-direction, the values vary linearly

106 5.3 Extended multiscale ﬁnite element method for the analysis of linear elastic heterogeneous materials

Figure 5.1: [54]The construction of the numerical basis functions

x x Figure 5.2: Displacement ﬁeld of T for the basis function Φp1x (left) and Φp1y (right) along each side. We ignore the displacement values in y-axis direction and remaining x displacement values in x-axis direction to establish the basis function Φp1 (x) = Φp1x y The construction of Φp1 (y) = Φp1y is similar to that of Φp1 (x), we have obtained the basis functions Φp1 (x) and Φp1 (y) Figure 5.2, the rest of the basis functions of the coarse element can be constructed in the similar way.

A modification of the construction of the basis functions is developed in [55] and used for small deformation elasto-plastic analysis of periodic truss materials. So that the multiscale basis functions for the displacement fields can be constructed in a more accurate way. The most key point is that the additional coupling terms of the basis functions are introduced in the improved construction method. For the element T Figure 5.1 (right), the displacements at all boundary nodes are constraint in y direction, at the same time the nodes on boundary 34 and boundary 23 are constraint in x- direction, a unit displacement is applied on node 1 in the positive x-direction and the values vary linearly along boundaries 12 and 14,the internal displacement field of the element can be obtained directly by standard finite element analysis in fine scale mesh x x and the basis functions Φp1 (x) and Φp1 (y) can be obtained, thus the basis functions x x x Φp1 = {Φp1x, Φp1y}.

x Here Φpiy is a coupled additional term and means that the displacement ﬁeld in y- direction within the element induced by unit displacement of node i in the x-direction.

The rest of the basis functions can be constructed in the similar way and the ﬁnal basis x x y y functions ie.Φpix, Φpiy, Φpiy and Φpix for i = 1, 2, 3, 4 obtained by this way

107 5 Extended Multiscale Finite Element Method in Linear Elasticity

Figure 5.3: [54] Schematic description of the EMsFEM

5.3.2 Stiﬀness matrix on the coarse mesh

The displacement vector of the nodes in the ﬁne scale omits the representation

uh = RuH (5.22) where uH is the displacement vector of nodes in the coarse mesh and R is the basis function matrix which contains the coeﬃcient vectors, representing a coarse basis function in terms of the ﬁne scale basis .

T T T T R = [R1 R2 ··· Rn ] (5.23)

" x x x x x x x x # Φp1x Φp1y Φp2x Φp2y Φp3x Φp3y Φpiy Φpiy Ri = y y y y y y y y (5.24) Φp1x Φp1y Φp2x Φp2y Φp3x Φp3y Φpiy Φpiy i = 1, 2 . . . , n where n is the total number of the the ﬁne scale mesh within the sub-grids, using (5.22) and (5.24), we have

e E uh = GeuH (5.25) where   Re1 R  G =  e2  (5.26) e R   e3 

Re4 (5.27) is the transformation matrix between the displacement vectors of micro-scale nodes and macro-scale nodes and e is an arbitrary ﬁne-scale element within the coarse element show Figure 5.3.

108 5.4 Convergence of the multiscale ﬁnite element method

The stiﬀness matrix of the coarse element is given by

N H X ˜ ˜ T KE = Ke, Ke = Ge KeGe (5.28) e=1 where Ke is the element stiﬀness matrix.

The global stiﬀness matrix on the coarse mesh is obtained as follows

M H K = Ai=1KE (i) (5.29)

M where Ai=1 is a matrix assembled operator and M is the total number of the coarse element.

The corresponding multiscale coarse space is conforming , we deﬁne the coarse space

Ms Ms p ¯ 1 VH := {Φp(k), x ∈ ΣH , k = 1, 2} ⊆ H0(Ω) (5.30)

Ms 1 Using the defined space of multiscale functions VH ⊆ H0(Ω) in the finite element Ms problem gives a multiscale finite element approximation uH which satisfies for all Ms Ms vH ∈ VH Z Ms Ms Ms Ms aΩ(uH , vH ) = σ(uH ): ε(vH ) dx Ω Z Ms = F · vH dx (5.31) Ω

It holds that Z Z Ms Ms X Ms Ms σ(uH ): ε(vH ) dx = σ(uH ): ε(vH ) dx Ω T T ∈TH

5.4 Convergence of the multiscale ﬁnite element method

In this section, the convergence of the MsFEM is presented. The MsFEM is deﬁned for non-periodic problem but the convergence analysis is made in the periodic case. As in [26],[39], we restrict ourselves to a periodic case.

We consider the following elasticity problem

Lu = f in Ω

u = 0 on ∂Ω (5.32)

109 5 Extended Multiscale Finite Element Method in Linear Elasticity

x x L is the elasticity operator, L = div(C( ): ε(·)), and C(x) = (Cijkl( )) is the fourth order elasticity tensor, which satisﬁes symmetry and positiv deﬁniteness. There exists α > 0 such that x C (y)ε ε ≥ αε ε ∀ y = , ∀ ε symmetric ijkl ij kl ij ij ij

The multiscale basis functions satisfy

Ms LΦj(k) = 0 in T ∈ TH . Recall that H is the coarse mesh size.

We refer to [39] for the convergence analysis of MsFEM, where the convergence for H < and H > cases are presented for a Dirichlet problem.

The convergence rate of MsFEM contains a term H (see [39]), the error becomes large when the two scale are close, when H ≈ the multiscale method attains a large error in H1 and L2 norms (see [26]).

The multiscale basis functions are smooth if H < and can be well approximated by the standard continuous linear (bilinear) basis functions, we apply the traditional ﬁnite element method analysis to the multiscale method. When H > the multiscale basis functions contains a smooth part and an oscillatory part, which cannot be approximated by linear ( bilinear ) functions, the MsFEM gives a convergence result uniform in as tends to zero, while the traditional FEM with piecewise polynomial basis functions does not.

Ms Theorem 5.1. [39](Convergence for H < ) Let u and uH be the solutions of (5.32) and (5.31), respectively. Then there exists a constant C, independent of H and , such that

Ms H ku − u k 1 ≤ C( )kfk 2 (5.33) H H (Ω) L (Ω) Ms H 2 ku − u k 2 ≤ C( ) kfk 2 (5.34) H L (Ω) L (Ω) Remark 5.1. Details of the proof for a Dirichlet problem can be found in [26],[39]. The proof for the transmission problem similarly follows from a convergence analysis of the standard finite element / boundary element coupling, using from [26],[39] that the multiscale basis functions do not differ significantly from piecewise linear hat functions. Remark 5.2. For H < the multiscale method gives the same rate of convergence as the linear finite element method , but this estimate is insufficient for practical applications, the estimates (5.33) and (5.34) blows up as → 0.

Ms Ms Let IH be the standard interpolation operator , and IH : C(Ω) −→ VH be the nodal interpolation operator deﬁned in [20] by

Ms X Ms IH u = u(xj)Φj (5.35) j

110 5.4 Convergence of the multiscale ﬁnite element method

From the deﬁnition of the multiscale basis functions, we have

Ms Ms L(IH u) = 0 in T, IH u = IH u on ∂T (5.36)

From the approximation theory, we have

ku − IH ukL2(T ) ≤ C1H|u|H1(T ) (5.37)

|u − IH u|H1(T ) ≤ C2|u|H1(T ) (5.38)

1 Lemma 5.1. Let u ∈ H (Ω) be the solution of (5.32). There exist constants C1 and C2 > 0 independent of H, such that

Ms 2 ku − IH ukL2(T ) = C1(H|u|H1(T )+H kfkL2(T )) (5.39) Ms |u − IH u|H1(T ) = C2(|u|H1(T )+HkfkL2(T )) (5.40)

Ms Ms 1 Proof. We refer to [26],[39], since IH u = IH u on ∂T , we have IH u − IH u ∈ H0(T ), and by the Poincar´e-Friedrichs inequality we obtain

Ms Ms kIH u − IH ukL2(T )≤ CH|IH u − IH u|H1(T )

From (5.36), we have Z x Ms MS Ms C( )ε(IH u ): ε(IH u − IH u) dx = 0, (5.41) T and similar to the proof of Lemma 6.3 in [26] we obtain Z Ms 2 x Ms Ms |IH u − IH u|H1(T )= C( )ε(IH u − IH u): ε(IH u − IH u) dx T Z Z x Ms x Ms = C( )ε(u − IH u): ε(IH u − IH u) dx − C( )ε(u): ε(IH u − IH u) dx T T Z Z x Ms x Ms = C( )ε(u − IH u): ε(IH u − IH u) dx − C( )f(IH u − IH u) dx T T Ms Ms ≤ C|u − IH u|H1(T )|IH u − IH u|H1(T )+kIH u − IH ukL2(T )kfkL2(T )

Ms ≤ C|IH u − IH u|H1(T )(|u − IH u|H1(T )+HkfkL2(T )) (5.42)

We obtain

Ms |IH u − IH u|H1(T ) ≤ C(|u|H1(T )+HkfkL2(T )) (5.43) Ms 2 kIH u − IH ukL2(T ) ≤ C(H|u|H1(T )+H kfkL2(T )) (5.44)

111 5 Extended Multiscale Finite Element Method in Linear Elasticity

Using (5.37), (5.38) ,(5.43) and (5.44), we get

Ms Ms |u − IH u|H1(T ) ≤ |u − IH u|H1(T )+|IH u − IH u|H1(T )

≤ C2(|u|H1(T )+HkfkL2(T )) Ms Ms ku − IH ukL2(T ) ≤ ku − IH ukL2(T )+kIH u − IH ukL2(T ) 2 ≤ C1(H|u|H1(T )+H kfkL2(T )

Now we discuss the convergence analysis for H > by exploring the asymptotic behavior Ms of both u and Φj . We consider the expansion of u:

u = u0 + u1 + ... (5.45) where u0 is the solution of the homogenized equation For more details (see [26],[39]).

0 Ms Let uI be the interpolant of u , using the multiscale basis functions Φj , note that uI Ms is diﬀerent from the deﬁnition of IH u. In the literature the following homogenization estimates are only available for the Dirichlet [39] and Neumann [49] problems, not the transmission problem.

Lemma 5.2. Let u be the solution of (5.32) and uI the interpolant of the homogenized 0 Ms solution u , using the multiscale basis functions Φj . Then there exist constants C1 and C2, independent of and H, such that 1 ku − u k 1 ≤ C Hkfk 2 +C ( ) 2 (5.46) I H (Ω) 1 L (Ω) 2 H 2 1 ku − u k 2 ≤ C H kfk 2 +C ( ) 2 (5.47) I L (Ω) 1 L (Ω) 2 H Ms Theorem 5.2. (Convergence for H > ) Let u and uH be the solutions of (5.32) and (5.31), respectively. Then there exist constants C1 and C2, independent of and H, such that

Ms 1 ku − u k 1 ≤ C Hkfk 2 +C ( ) 2 (5.48) H H (Ω) 1 L (Ω) 2 H (5.49)

It is shown in [39] that

Ms 2 ku − u k 2 = O(H + ) (5.50) H L (Ω) H Remark 5.3. While we expect these results to hold for the transmission problem, an analysis of the homogenized transmission problem has not been reported. We leave the necessary analysis as an open problem. Remark 5.4. For H > the multiscale method converges to the homogenized solution u0 in the limit as → 0.

112 5.5 A posteriori error for multiscale ﬁnite element method

5.5 A posteriori error for multiscale ﬁnite element method

We consider the problem

Lu = f in Ω (5.51)

Ms Lemma 5.3. Let u be the solution of (5.51) and uH be the solution of the discrete problem (5.31). Then there holds the estimate: Z Ms X Ms a(eH , v) = f(v − vH ) dx T T ∈TH 4 Z X X Ms Ms + [σ(uH ) · n]El (v − vH ) ds (5.52) El⊂∂T T ∈TH l=1

Ms Ms where eH = u − uH .

Ms Ms Proof. We consider the quantity eH = u − uH , called multiscale discretization error, in the following we derive and analyze an upper bound for the quantity q Ms Ms Ms keH kΩ:= a(eH , eH ) (5.53) measured in the energy norm.

Ms Ms Given the multiscale solution uH ∈ VH of the discrete problem (5.31) , we obtain the so called error residual equation

Ms Ms Ms a(eH , v) = a(u − uH , v) = F (v) − a(uH , v) (5.54)

Recalling to

Ms 1 aT (Φp(k), v) = 0 for all v ∈ H0(T ),T ⊂ wp (5.55) and Ms Ms p ¯ 1 VH := {Φp(k), x ∈ ΣH , k = 1, 2} ⊆ H0(Ω) we obtain

Ms 1 aT (uH , v) = 0 for all v ∈ H0(T ) (5.56)

Ms Ms Let v ∈ V and vH ∈ VH , using this we rewrite the discrete problem (5.31) as follows

Ms Ms Ms 0 = F (vH ) − a(uH , vH ) (5.57)

113 5 Extended Multiscale Finite Element Method in Linear Elasticity

Ms Ms with the discretization error eH = u−uH the Galerkin-method for ﬁnite sub-domains yields the important Galerkin orthogonality

Ms Ms Ms Ms a(eH , vH ) = 0 ∀vH ∈ VH (5.58)

Subtracting (5.54) from (5.57) we obtain Z Z Ms X Ms Ms a(eH , v) = f(v − vH ) dx + t(v − vH ) ds T ΓN ∩∂T T ∈TH Z X Ms Ms − σ(uH ): ε(v − vH ) dx ∀ v ∈ V (5.59) T T ∈TH

R Ms Ms integration by parts of the term T σ(uH ): ε(v − v ) dx in (5.59) we obtain Z Z Ms X Ms Ms a(eH , v) = f(v − vH ) dx + t(v − vH ) ds T ΓN ∩∂T T ∈TH Z I X Ms Ms Ms Ms + div σ(uH )(v − vH ) dx − σ(uH ) · n · (v − vH ) ds T ∂T T ∈TH (5.60)

For ﬁxed T ∈ TH let El be an edge of T with l = 1, 2, 3, 4, we introduce the jump of tractions at the element boundaries

 1 + Ms + − Ms − σ (u ) · n + σ (u ) · n if El ⊂ ΓT = ∂T  2 H El H El   Ms  [σ(u ) · n]E := Ms H l t − σ(uH ) · nEl if El ⊂ ΓT,N = ∂T ∩ ΓN     0 if El ⊂ ΓT,D = ∂T ∩ ΓD (5.61)

here nEl is the normal on El

Ms Recall that by contruction of the multiscale basis functions, we have divσ(uH ) = 0 in T , we can now regroup the boundary terms in(5.60) for every ﬁxed T so that (5.60) becomes Z Ms X Ms a(eH , v) = f(v − vH ) dx T T ∈TH 4 Z X X Ms Ms + [σ(uH ) · n]El (v − vH ) ds El⊂∂T T ∈TH l=1 (5.62)

In the following, we derive a-posteriori error estimates for H < and H > .

114 5.5 A posteriori error for multiscale ﬁnite element method

Theorem 5.3. (A posteriori error estimate for H < )

Ms Let u be the solution of (5.32) and uH be the solution of the discrete problem (5.31). Then there holds the estimate:

X X X X a(eMs, eMs) η2 + η2 + η2 + η2 (5.63) H H . Th TH Eh EH T ⊆Ω T ⊆Ω ˚ ˚ h H Eh⊆EH ⊆Ω EH ⊆Ω where

X X Z η2 = H2(T ) |f − f|2 dx Th h Th Th⊆Ω Th⊆Ω X X η2 = H4|f|2 TH TH ⊆Ω TH ⊆Ω X X Z η2 = H(E ) ([σ(uMs) · n] − [σ(uMs) · n])2 Eh h H H ˚ ˚ Eh Eh⊆EH ⊆Ω Eh⊆EH ⊆Ω X X η2 = H2([σ(uMs) · n])2 EH H

EH ⊆˚Ω EH ⊆˚Ω Z Z 1 Ms 1 Ms f(x) = f dx, [σ(uH ) · n](x) = [σ(uH ) · n] dx |TH | TH |EH | EH

Proof. From Lemma 5.3, we have

Z Ms X Ms a(eH , v) = f(v − vH ) dx T T ∈TH 4 Z X X Ms Ms + [σ(uH ) · n]El (v − vH ) ds (5.64) El⊂∂T T ∈TH l=1

First, we estimate the term

Z X Ms A1 := f(v − vH ) dx (5.65) T T ∈TH

1 R Let f(x) = f, if x ∈ TH then |TH | TH

Z Z Z X Z Z |f|2 = (f − f + f)2 = (f − f)2 + 2 f (f − f) + (f)2 Ω Ω Ω TH Ω TH ∈TH

However R (f − f) = 0 TH Z Z Z |f|2 = (f − f)2 + (f)2 (5.66) Ω Ω Ω

115 5 Extended Multiscale Finite Element Method in Linear Elasticity

Ms Ms Ms Ms Using (5.39), choosing v = eH and vH = IH eH , we obtain

X Ms Ms Ms A ≤ kfk 2 ke − I e k 2 1 L (TH ) H H H L (TH ) TH ⊆Ω X Ms 2 ≤ C kfk 2 (H|e | 1 + H kfk 2 ) L (TH ) H H (TH ) L (TH ) TH ⊆Ω X 2 2 X Ms ≤ C( H kfk 2 + Hkfk 2 |e | 1 L (TH ) L (TH ) H H (TH ) TH ⊆Ω TH ⊆Ω 0 X 2 2 1 Ms 2 ≤ C H kfk 2 + |eH | 1 (5.67) L (TH ) 2 H (TH ) TH ⊆Ω P 2 2 Now we estimate the term H kfk 2 , employing (5.66), we get TH ⊆Ω L (TH ) Z X 2 2 X 2 X 2 H kfk 2 = H ( |f| dx) L (TH ) Th TH ⊆Ω TH ⊆Ω Th⊆TH X X Z X Z = H2{ |f − f|2 dx + |f|2 dx} Th Th TH ⊆Ω Th⊆TH Th⊆TH Z X 2 X 2 X 2 = H { |f − f| dx + |Th||f| } Th TH ⊆Ω Th⊆TH Th⊆TH Z X 2 X 2 2 = H { |f − f| dx+|TH ||f| } Th TH ⊆Ω Th⊆TH Z X 2 2 X 4 2 = H (Th) |f − f| dx + H |f| Th Th⊆Ω TH ⊆Ω X X = η2 + η2 . (5.68) Th TH Th⊆Ω TH ⊆Ω

2 Here f|TH is a constant on TH , and |TH | ∼ H . We obtain

X 2 X 2 1 Ms 2 A1 . ηT + ηT + |eH | 1 (5.69) h H 2 H (TH ) Th⊆Ω TH ⊆Ω

Now we estimate the term Z X Ms Ms A2 := [σ(uH ) · n]EH (v − vH ) ds. (5.70) EH EH ⊆˚Ω where EH is an interior edge of T.

Ms Ms Ms MS Ms Ms Ms Choosing v = eH and vH = IH eH , since IH eH = IH eH on ∂T , we get

Ms Ms Ms Ms Ms Ms kv − v k 2 = ke − I e k 2 = ke − I e k 2 . (5.71) H L (EH ) H H H L (EH ) H H H L (EH ) 1 From [[21],Lemma 4], for every edge EH of TH and u ∈ H (TH ), we have

2 2 2 2 Hkuk 2 ≤ C(kuk 2 +H |u| 1 ) (5.72) L (EH ) L (TH ) H (TH )

116 5.5 A posteriori error for multiscale ﬁnite element method

Using (5.72), we get

Ms 2 Ms Ms Ms 2 kv − v k 2 = ke − I e k 2 H L (EH ) H H H L (EH ) C Ms Ms 2 2 Ms Ms 2 ≤ keH − IH eH k 2 +H |eH − IH eH | 1 H L (TH ) H (TH ) C 2 Ms 2 2 Ms 2 ≤ H |eH | 1 + H |eH | 1 H H (TH ) H (TH ) 0 Ms 2 ≤ C H|e | 1 . (5.73) H H (TH ) Using (5.73), we obtain with Young’s inequality

X Ms Ms A ≤ k[σ(u ) · n]k 2 kv − v k 2 2 H L (EH ) H L (EH )

EH ⊆˚Ω Ms X 1 Ms ≤|e | 1 H 2 k[σ(u ) · n]k 2 H H (Ω) H L (EH )

EH ⊆˚Ω Z X Ms 2 1 Ms 2 ≤ C H ([σ(uH ) · n]) ds + |eH |H1(Ω). (5.74) EH 2 EH ⊆˚Ω Now we estimate the ﬁrst term in (5.74), with

Z Ms 1 Ms [σ(uH ) · n](x) = [σ(uH ) · n] (5.75) |EH | EH we have Z Z X Ms 2 X X Ms 2 A22 := H ([σ(uH ) · n]) = H ([σ(uH ) · n]) EH E ⊆E Eh EH ⊆˚Ω EH ⊆˚Ω h H   Z Z X X Ms Ms 2 X Ms 2 = H  ([σ(uH ) · n] − [σ(uH ) · n]) + ([σ(uH ) · n])  E ⊆E Eh E ⊆E Eh EH ⊆˚Ω h H h H   Z X X Ms Ms 2 X Ms 2 = H  ([σ(uH ) · n] − [σ(uH ) · n]) + |Eh|([σ(uH ) · n])  E ⊆E Eh E ⊆E EH ⊆˚Ω h H h H   Z X X Ms Ms 2 Ms 2 = H  ([σ(uH ) · n] − [σ(uH ) · n]) +|EH |([σ(uH ) · n])  E ⊆E Eh EH ⊆˚Ω h H Z X Ms Ms 2 X 2 Ms 2 = H(Eh) ([σ(uH ) · n] − [σ(uH ) · n]) + H ([σ(uH ) · n]) ˚ Eh ˚ Eh⊆EH ⊆Ω EH ⊆Ω X X = η2 + η2 . (5.76) Eh EH ˚ ˚ Eh⊆EH ⊆Ω EH ⊆Ω where |EH | ∼ H.

We obtain

X 2 X 2 1 Ms 2 A2 η + η + |e | 1 . (5.77) . Eh EH 2 H H (Ω) ˚ ˚ Eh⊆EH ⊆Ω EH ⊆Ω

117 5 Extended Multiscale Finite Element Method in Linear Elasticity

We throw the term 1 |eMs|2 to the left hand side, the estimate of the Theorem 2 H H1(Ω) follows immediately.

Theorem 5.4. (A posteriori error estimate for H > )

Ms Let u be the solution of (5.32) and uH be the solution of the discrete problem (5.31). Then there holds the estimate: X X X X a(eMs, eMs) η2 + η2 + η2 + η2 H H . Th TH Eh EH T ⊆Ω T ⊆Ω ˚ ˚ h H Eh⊆EH ⊆Ω EH ⊆Ω 1   2 X 2 +  kfk 2 ( ) (5.78) L (TH ) H TH ⊆Ω

Proof. To prove the theorem, we ﬁrst denote

Ms X Ms uI (x) = IH u0(x) = u0(xj)Φj , (5.79) j where u0 is the solution of the homogenized equation and uI is the interpolant of u0 Ms using the multiscale basis functions Φj , note that uI is diﬀerent from the deﬁnition Ms of IH u. We know that

L(uI ) = 0 in T, uI = IH u0 on ∂T (5.80)

First, we estimate the term Z X Ms A1 = f(v − vH ) dx (5.81) T T ∈TH

Ms Ms Ms Choosing v = eH and vH = uI − uH , using (5.47) and Cauchy Schwarz inequality, we obtain X A ≤ kfk 2 ku − u k 2 1 L (TH ) I L (TH ) TH ⊆Ω X 2 1 ≤ kfk 2 (C1H kfk 2 +C2( ) 2 ) L (TH ) L (Ω) H TH ⊆Ω X 2 2 X 1 ≤ C1H kfk 2 + C2kfk 2 ( ) 2 L (TH ) L (TH ) H TH ⊆Ω TH ⊆Ω (5.82)

From (5.68), we obtain

1   2 X 2 X 2 X 2 A1 . ηT + ηT +  kfk 2 ( ) (5.83) h H L (TH ) H Th⊆Ω TH ⊆Ω TH ⊆Ω

118 5.6 Coupling FEM-BEM

Now we estimate the term Z X Ms Ms A2 = [σ(uH ) · n]EH (v − vH ) ds. (5.84) EH EH ⊆˚Ω

Ms Ms Ms Choosing v = eH and vH = IH (u − uH ) on ∂T , we obtain (see (5.72)-(5.77))

X 2 X 2 1 Ms 2 A2 η + η + |e | 1 . (5.85) . Eh EH 2 H H (Ω) ˚ ˚ Eh⊆EH ⊆Ω EH ⊆Ω

We throw the term 1 |eMs|2 to the left hand side, the estimate of the theorem 2 H H1(Ω) follows immediatly.

5.6 Coupling FEM-BEM

Let Ω be a bounded Lipschitz domain with boundary Γ := ∂Ω and exterior domain 2 Ωc := R \ Ω. The displacement ﬁeld u of the body satisﬁes the elasticity material behaviour

σ(u) = C : ε(u), (5.86) where σ is the stress tensor, the strain tensor ε is given by the symmetric part of the deformation gradient. 1 ε(u) = (∇u + ∇uT ). (5.87) 2 Given a volume force f the equilibrium equation reads

div σ(u) + f = 0 in Ω. (5.88)

The exterior problem consists of the Navier-Lam´eequation

∗ 0 = −4 uc := −µ24uc − (λ2 + µ2)grad div u in Ωc, (5.89) and a radiation condition of the form

α −1−α D (uc − a)(x) = O(|x| ), α = 0, 1, (|x| → ∞), (5.90)

2 where D = ∂/∂xj and a ∈ R is a constant vector.

1 − 1 The transmission problem has the data f ∈ L2(Ω), u0 ∈ H 2 (Γ), and t0 ∈ H 2 (Γ) and 1 1 consists in ﬁnding (u, uc) ∈ H (Ω) × Hloc(Ωc) satisfying (5.88), (5.89), (5.90), and the interface conditions:

u − uc = u0 on Γ (5.91)

∗ σ(u) · n = T (uc) + t0 on Γ (5.92)

119 5 Extended Multiscale Finite Element Method in Linear Elasticity where T ∗ is the conormal derivative related to the Lam´eoperator 4∗.

∗ T (uc) := 2µ2∂nuc + λ2n div uc + µ2n × curluc (5.93)

Here, ∂n is the normal derivative and n is the unit normal vector on Γ pointing from Ω into Ωc.

1 ∗ The Cauchy data of uc ∈ Hloc(Ωc) with 4 uc = 0 satisfy

∗ 1 − 1 (ξ, φ) := (uc|Γ,T (uc)|Γ) ∈ H 2 (Γ) × H 2 (Γ) (5.94)

The Steklov-Poincar´eoperator for the exterior Lam´eproblem is given as

0 −1 S := (W + (K − 1)V (K − 1))2, (5.95) where W = 2W , K0 = 2K 0 , V = 2V , K = 2K in Chapter2 which satisﬁes

∗ T uc|Γ = Suc|Γ (5.96)

5.6.1 The weak formulation of the Model Problem

The weak form of the interface problem (5.88)-(5.92) reads as:

Find u ∈ H1(Ω), such that ∀ v ∈ H1(Ω)

hσ(u), ε(v)i + hSu|Γ, v|Γi = hf, vi + ht0 + Su0, v|Γi (5.97)

Let

1 1 H 2 (Γ)/R := {φ ∈ H 2 (Γ), h1, φi = 0} (5.98)

We consider the following weak formulation for a multiscale problem ( cf. [17]):

1 − 1 1 Find (u, ξ, φ) ∈ H (Ω) × H 2 (Γ) × H 2 (Γ)/R, such that

1 hCε(u), ε(v)i − hφ, vi = hf, vi + ht0, vi ∀v ∈ H (Ω) (5.99a)

1 1 − 1 −hu, ψi − hVφ, ψi + h(K + 1)ξ, ψi = −hu , ψi ∀ψ ∈ H 2 (Γ) (5.99b) 2 2 0 1 0 1 h(K + 1)φ, θi + hWξ, θi = 0 ∀θ ∈ H 2 (Γ)/ (5.99c) 2 R 1 With the solution u ∈ H (Ω), we deﬁne φ := S(u0 − u|Γ) and ξ := u|Γ − u0.

We rewrite (5.99) as follows:

1 − 1 1 Find (u, ξ, φ) ∈ H (Ω) × H 2 (Γ) × H 2 (Γ)/R, such that

1 − 1 1 B(u, ξ, φ; v, ψ, θ) = L(v, ψ) ∀(v, ψ, θ) ∈ H (Ω) × H 2 (Γ) × H 2 (Γ)/R (5.100)

120 5.6 Coupling FEM-BEM

5.6.2 The discrete problem for Ms-FEM-BEM

− 1 1 1 1 Ms 2 2 − 2 2 Let VH ×HH ×HH be a family of ﬁnite dimensional subspaces of V×H ×H (Γ)/R, Ms where VH is the deﬁned space of multiscale functions. The discretized version of (5.100) reads as:

1 1 Ms Ms Ms Ms Ms − 2 2 B(uH , φH , ξH ; vH , ψH , θH ) = L(vH , ψH ) ∀(vH , ψH , θH ) ∈ VH × HH × HH (5.101)

As in [17], we consider the relation:

1 1 hφ − φ , u − uMsi = − hW(ξ − ξ ), ξ − ξ i − hV(φ − φ ), φ − φ i H H Γ 2 H H 2 H H 1 + hu − uMs, φ − φ˜ i + hV(φ − φ ), φ − φ˜ i H H 2 H H 1 1 0 − h(K + 1)(ξ − ξ ), φ − φ˜ i + h(K + 1)(φ − φ ), ξ − ξ˜ i 2 H H 2 H H 1 + hW(ξ − ξ ), ξ − ξ˜ i (5.102) 2 H H

− 1 1 ˜ 2 ˜ 2 for all φH ∈ HH , ξH ∈ HH

5.6.3 A priori error estimate

Using the relation (5.102) and Cauchy-Schwarz inequality, we obtain

Ms Ms Ms Ms B(u − uH , φ − φH , ξ − ξH ; u − uH , φ − φH , ξ − ξH ) = hCε(u − uH ), ε(u − uH )i 1 − hφ − φ , u − uMsi − hu − uMs, φ − φ i − hV(φ − φ ), φ − φ i H H Γ H H 2 H H 1 1 0 1 + h(K + 1)(ξ − ξ ), φ − φ i − h(K + 1)(φ − φ ), ξ − ξ i − hW(ξ − ξ ), ξ − ξ i 2 H H 2 H H 2 H H 1 1 = hC ε(u − uMs), ε(u − uMs)i + hW(ξ − ξ ), ξ − ξ i + hV(φ − φ ), φ − φ i − 2A H H 2 H H 2 H H 1 2 Ms 2 2 2 & kC ε(u − uH )kL (Ω)+kξ − ξH k 1 +kφ − φH k − 1 −2A (5.103) 2 H 2 (Γ)/R H 2 (Γ) where 1 1 A := − hW(ξ − ξ ), ξ − ξ i − hV(φ − φ ), φ − φ i 2 H H 2 H H 1 + hu − uMs, φ − φ˜ i + hV(φ − φ ), φ − φ˜ i H H 2 H H 1 1 0 − h(K + 1)(ξ − ξ ), φ − φ˜ i + h(K + 1)(φ − φ ), ξ − ξ˜ i 2 H H 2 H H 1 + hW(ξ − ξ ), ξ − ξ˜ i (5.104) 2 H H

121 5 Extended Multiscale Finite Element Method in Linear Elasticity

1 1 Ms Ms − 2 2 On the other hand , for all (vH , ψH , θH ) ∈ VH × HH × HH , we have

Ms Ms B(u − uH , φ − φH , ξ − ξH ; u − uH , φ − φH , ξ − ξH ) = Ms Ms B(u − uH , φ − φH , ξ − ξH ; u − vH , φ − ψH , ξ − θH ) Ms Ms Ms + B(u − uH , φ − φH , ξ − ξH ; vH − uH , ψH − φH , θH − ξH ) (5.105)

Ms Ms Ms Since B(u − uH , φ − φH , ξ − ξH ; vH − uH , ψH − φH , θH − ξH ) = 0, we obtain

1 2 Ms 2 2 2 kC ε(u − uH )kL (Ω)+kξ − ξH k 1 +kφ − φH k − 1 2 H 2 (Γ)/R H 2 (Γ) Ms Ms . B(u − uH , φ − φH , ξ − ξH ; u − vH , φ − ψH , ξ − θH ) + 2A Ms Ms Ms = hCε(u − uH ), ε(u − vH )i − hφ − φH , u − vH iΓ 1 − hu − uMs, φ − ψ i − hV(φ − φ ), φ − ψ i H H 2 H H 1 1 0 1 + h(K + 1)(ξ − ξ ), φ − ψ i − h(K + 1)(φ − φ ), ξ − θ i − hW(ξ − ξ ), ξ − θ i 2 H H 2 H H 2 H H Ms ˜ ˜ + 2hu − uH , φ − φH i + hV(φ − φH ), φ − φH i ˜ 0 ˜ ˜ − h(K + 1)(ξ − ξH ), φ − φH i + h(K + 1)(φ − φH ), ξ − ξH i + hW(ξ − ξH ), ξ − ξH i Ms Ms Ms = hCε(u − uH ), ε(u − vH )i − hφ − φH , u − vH iΓ 1 1 + hu − uMs, φ − ψ − 2φ˜ i + hV(φ − φ ), φ − ψ − φ˜ i H H H H 2 2 H H 1 1 0 1 1 + h(K + 1)(ξ − ξ ), φ˜ − ψ − φi + h(K + 1)(φ − φ ), ξ − θ − ξ˜ i H H 2 H 2 H 2 2 H H 1 1 + hW(ξ − ξ ), ξ + θ − ξ˜ i H 2 2 H H 1 2 Ms 1 1 . kC ε(u − uH )kL2(Ω)+kξ − ξH k +kφ − φH k − · H 2 (Γ)/R H 2 (Γ) 1 2 Ms kC ε(u − vH )kL (Ω)+kξ − θH k 1 +kφ − ψH k − 1 2 H 2 (Γ)/R H 2 (Γ) (5.106)

˜ ˜ After choosing θH = ξH , ψH = φH , we get the following theorem.

Theorem 5.5.

1 2 Ms 2 2 2 kC ε(u − uH )kL (Ω)+kξ − ξH k 1 +kφ − φH k − 1 2 H 2 (Γ)/R H 2 (Γ) 1 2 Ms inf kC ε(u − vH )kL (Ω)+kξ − θH k 1 +kφ − ψH k − 1 . 1 1 2 H 2 (Γ)/ H 2 (Γ) Ms Ms − 2 2 R (vH ,ψH ,θH )∈VH ×HH ×HH (5.107)

5.6.4 A posteriori error

In this section, we derive a-posteriori error estimates for H < and H > .

122 5.6 Coupling FEM-BEM

Theorem 5.6. (A posteriori error estimate for H < )

Ms Let (u, φ, ξ) be the solution of (5.100) and (uH , φH , ξH ) be the solution of the discrete problem (5.101). Then there holds the estimate:

1 2 Ms 2 2 2 kC ε(u − uH )kL (Ω)+kξ − ξH k 1 +kφ − φH k − 1 2 H 2 (Γ)/R H 2 (Γ) X Ms 2 Hkσ(u ) · n − t0 − φH k 2 . H L (EH ) EH ⊆Γ X ∂ Ms 1 1 2 + Hk (u0 − uH + V φH − (K + 1)ξH )k 2 ∂s 2 2 L (EH ) EH ⊆Γ X 0 2 + Hk(K + 1)φH + W ξH k 2 L (EH ) EH ⊆Γ X X X X + η2 + η2 + η2 + η2 Th TH Eh EH T ⊆Ω T ⊆Ω ˚ ˚ h H Eh⊆EH ⊆Ω EH ⊆Ω

X X Z X X η2 = H2(T ) |f − f|2 dx, η2 = H4|f|2 Th h TH Th Th⊆Ω Th⊆Ω TH ⊆Ω TH ⊆Ω X X Z η2 = H(E ) ([σ(uMs) · n] − [σ(uMs) · n])2 Eh h H H ˚ ˚ Eh Eh⊆EH ⊆Ω Eh⊆EH ⊆Ω X X η2 = H2([σ(uMs) · n])2 EH H

EH ⊆˚Ω EH ⊆˚Ω Z Z 1 Ms 1 Ms f(x) = f dx, [σ(uH ) · n](x) = [σ(uH ) · n] dx |TH | TH |EH | EH

Proof. Recall that

1 2 Ms 2 2 2 kC ε(u − uH )kL (Ω)+kξ − ξH k 1 +kφ − φH k − 1 2 H 2 (Γ)/R H 2 (Γ) Ms Ms . B(u − uH , φ − φH , ξ − ξH ; u − vH , φ − ψH , ξ − θH ) + 2A Ms Ms Ms = L(u − vH , φ − ψH ) − B(u , φH , ξH ; u − vH , φ − ψH , ξ − θH ) 1 1 + 2hu − uMs + V(φ − φ ) − (K + 1)(ξ − ξ ), φ − φ˜ i 0 H 2 H 2 H H 0 ˜ + h(K + 1)(φ − φH ) + W(ξ − ξH ), ξ − ξH i Z Ms Ms = f(u − vH ) dx + ht0, u − vH i − hu0, φ − ψH i Ω Ms Ms Ms Ms − hCε(uH ), ε(u − vH )i + hφH , u − vH i + huH , φ − ψH i 1 1 + hVφ , φ − ψ i − h(K + 1)ξ , φ − ψ i 2 H H 2 H H 1 0 1 + h(K + 1)φ , ξ − θ i + hWξ , ξ − θ i 2 H H 2 H H 1 1 + 2hu − uMs + V(φ − φ ) − (K + 1)(ξ − ξ ), φ − φ˜ i 0 H 2 H 2 H H 0 ˜ + h(K + 1)(φ − φH ) + W(ξ − ξH ), ξ − ξH i (5.108)

123 5 Extended Multiscale Finite Element Method in Linear Elasticity

We estimate the last term in (5.108)

0 ˜ A1 := h(K + 1)(φ − φH ) + W(ξ − ξH ), ξ − ξH i (5.109)

Using (5.99c), Cauchy Schwarz and the inverse inequality, we get

0 ˜ A1 ≤ k(K + 1)φH + WξH k 1 kξ − ξH k 1 H− 2 (Γ) H 2 (Γ) 1 0 ˜ H 2 k(K + 1)φH + WξH kL2(Γ) infkξ − ξH k 1 (5.110) . ˜ H 2 (Γ) ξH

Similarly we have

1 0 1 A := h(K + 1)φ , ξ − θ i + hWξ , ξ − θ i 2 2 H H 2 H H 1 0 H 2 k(K + 1)φH + WξH k 2 infkξ − θH k 1 , (5.111) . L (Γ) 2 θH H (Γ)

Using (5.99b), Cauchy Schwarz and the inverse inequality, we get

1 1 A := hu − uMs + V(φ − φ ) − (K + 1)(ξ − ξ ), φ − φ˜ i 3 0 H 2 H 2 H H Ms 1 1 ˜ ku0 − uH − VφH + (K + 1)ξH k 1 infkφ − φH k − 1 . H 2 (Γ) ˜ H 2 (Γ) 2 2 φH 1 ∂ 1 1 2 Ms ˜ ≤ H k (u0 − uH + VφH − (K + 1)ξH )kL2(Γ) infkφ − φH k − 1 ˜ H 2 (Γ) ∂s 2 2 φH (5.112)

Similarly we have

1 1 A := huMs, φ − ψ i − hu , φ − ψ i + hVφ , φ − ψ i − h(K + 1)ξ , φ − ψ i 4 H H 0 H 2 H H 2 H H 1 ∂ Ms 1 1 ≤ H 2 k (u0 − u − VφH + (K + 1)ξH )k 2 infkφ − ψH k 1 H L (Γ) − 2 ∂s 2 2 ψH H (Γ) (5.113)

We estimate the interior terms, emploing the integration by part, we obtain Z Ms Ms Ms A5 = f(u − vH ) dx − hCε(uH ), ε(u − vH )i Ω Z Z X Ms Ms X Ms Ms = {f + divσ(uH )}(u − vH ) − [σ(uH ) · n]EH (u − vH ) ds T ⊆Ω TH EH H EH ⊆˚Ω Z X Ms Ms − σ(uH ) · n|EH (u − vH ) ds. (5.114) Eh EH ⊆Γ

Ms Recall that by contruction of the multiscale basis functions, we have divσ(uH ) = 0 in T .

124 5.6 Coupling FEM-BEM

Using (5.109)-(5.114), we obtain

1 2 Ms 2 2 2 kC ε(u − uH )kL (Ω)+kξ − ξH k 1 +kφ − φH k − 1 2 H 2 (Γ)/R H 2 (Γ) Z Z Ms X Ms Ms . f(u − vH ) dx + [σ(uH ) · n]EH (u − vH ) ds Ω EH EH ⊆˚Ω Ms Ms − hσ(uH ) · n − t0 − φH , u − vH i 1 0 ˜ + H 2 k(K + 1)φH + WξH kL2(Γ) infkξ − ξH k 1 ˜ H 2 (Γ) ξH 1 ∂ 1 1 2 Ms ˜ + H k (u0 − uH + VφH − (K + 1)ξH )kL2(Γ) infkφ − φH k − 1 ˜ H 2 (Γ) ∂s 2 2 φH (5.115)

First, we estimate the term

Z Ms B1 := f(u − vH ) dx (5.116) Ω

1 R Let f(x) = f, if x ∈ TH then |TH | TH

Z Z Z X Z Z |f|2 = (f − f + f)2 = (f − f)2 + 2 f (f − f) + (f)2 Ω Ω Ω TH Ω TH ∈TH

However R (f − f) = 0 TH

Z Z Z |f|2 = (f − f)2 + (f)2 (5.117) Ω Ω Ω

Ms Ms Ms Ms Ms Using (5.39), choosing vH = uH − IH (u − uH ), with e := u − uH , we obtain

X Ms B ≤ kfk 2 ku − v k 2 1 L (TH ) H L (TH ) TH ⊆Ω X 2 ≤ C kfk 2 (H|e| 1 + H kfk 2 ) L (TH ) H (TH ) L (TH ) TH ⊆Ω X 2 2 X ≤ C( H kfk 2 + Hkfk 2 |e| 1 L (TH ) L (TH ) H (TH ) TH ⊆Ω TH ⊆Ω 0 X 2 2 2 ≤ C H kfk 2 + |e| 1 (5.118) L (TH ) 2 H (TH ) TH ⊆Ω

125 5 Extended Multiscale Finite Element Method in Linear Elasticity

P 2 2 Now we estimate the term H kfk 2 , employing (5.117), we get TH ⊆Ω L (TH )

Z X 2 2 X 2 X 2 H kfk 2 = H ( |f| dx) L (TH ) Th TH ⊆Ω TH ⊆Ω Th⊆TH X X Z X Z = H2{ |f − f|2 dx + |f|2 dx} Th Th TH ⊆Ω Th⊆TH Th⊆TH Z X 2 X 2 X 2 = H { |f − f| dx + |Th||f| } Th TH ⊆Ω Th⊆TH Th⊆TH Z X 2 X 2 2 = H { |f − f| dx+|TH ||f| } Th TH ⊆Ω Th⊆TH Z X 2 2 X 4 2 = H (Th) |f − f| dx + H |f| Th Th⊆Ω TH ⊆Ω X X = η2 + η2 . (5.119) Th TH Th⊆Ω TH ⊆Ω

2 Hier f|TH is a constant on TH , and |TH | ∼ H . We obtain

X 2 X 2 2 B1 . ηT + ηT + |e| 1 (5.120) h H 2 H (TH ) Th⊆Ω TH ⊆Ω

Now we estimate the term Z X Ms Ms B2 = [σ(uH ) · n]EH (u − vH ) ds. (5.121) EH EH ⊆˚Ω

Ms Ms Ms Ms Ms Choosing vH = uH − IH (u − uH ), since IH u = IH u on ∂T , we get

Ms Ms ku − v k 2 = ke − I ek 2 = ke − I ek 2 . (5.122) H L (EH ) H L (EH ) H L (EH )

1 From [[21],Lemma 4], for every edge EH of TH and u ∈ H (TH ), we have

2 2 2 2 Hkuk 2 ≤ C(kuk 2 +H |u| 1 ) (5.123) L (EH ) L (TH ) H (TH )

Using (5.123), we get

Ms 2 Ms 2 C 2 2 2 ku − vH k 2 = ke − IH ek 2 ≤ ke − IH ek 2 +H |e − IH e| 1 L (EH ) L (EH ) H L (TH ) H (TH ) C 2 2 2 2 ≤ H |e| 1 + H |e| 1 H H (TH ) H (TH ) 0 2 ≤ C H|e| 1 . (5.124) H (TH )

126 5.6 Coupling FEM-BEM

Using (5.124), we obtain

X Ms Ms B ≤ k[σ(u ) · n]k 2 ku − v k 2 2 H L (EH ) H L (EH )

EH ⊆˚Ω X 1 Ms ≤|e| 1 H 2 k[σ(u ) · n]k 2 H (Ω) H L (EH )

EH ⊆˚Ω Z X Ms 2 2 ≤ C H ([σ(uH ) · n]) ds + |e|H1(Ω). (5.125) EH 2 EH ⊆˚Ω

Now we estimate the ﬁrst term in (5.125), with

Z Ms 1 Ms [σ(uH ) · n](x) = [σ(uH ) · n] (5.126) |EH | EH we have Z Z X Ms 2 X X Ms 2 B22 = H ([σ(uH ) · n]) = H ([σ(uH ) · n]) EH E ⊆E Eh EH ⊆˚Ω EH ⊆˚Ω h H   Z Z X X Ms Ms 2 X Ms 2 = H  ([σ(uH ) · n] − [σ(uH ) · n]) + ([σ(uH ) · n])  E ⊆E Eh E ⊆E Eh EH ⊆˚Ω h H h H   Z X X Ms Ms 2 X Ms 2 = H  ([σ(uH ) · n] − [σ(uH ) · n]) + |Eh|([σ(uH ) · n])  E ⊆E Eh E ⊆E EH ⊆˚Ω h H h H   Z X X Ms Ms 2 Ms 2 = H  ([σ(uH ) · n] − [σ(uH ) · n]) +|EH |([σ(uH ) · n])  E ⊆E Eh EH ⊆˚Ω h H Z X Ms Ms 2 X 2 Ms 2 = H(Eh) ([σ(uH ) · n] − [σ(uH ) · n]) + H ([σ(uH ) · n]) ˚ Eh ˚ Eh⊆EH ⊆Ω EH ⊆Ω X 2 X 2 2 = η + η + |e| 1 . (5.127) Eh EH 2 H (Ω) ˚ ˚ Eh⊆EH ⊆Ω EH ⊆Ω where |EH | ∼ H.

We obtain

X 2 X 2 2 B2 η + η + |e| 1 . (5.128) . Eh EH 2 H (Ω) ˚ ˚ Eh⊆EH ⊆Ω EH ⊆Ω

Now we estimate the term

Ms Ms B3 := hσ(uH ) · n − t0 − φH , u − vH i (5.129)

127 5 Extended Multiscale Finite Element Method in Linear Elasticity

Ms Ms Ms Ms Choosing vH = uH − IH (u − uH ),

Ms Ms B3 ≤ kσ(uH ) · n − t0 − φH kL2(Γ)ku − vH kL2(Γ) 1 Ms ≤ C|e|H1(Ω)H 2 kσ(uH ) · n − t0 − φH kL2(Γ) X Hkσ(uMs) · n − t − φ k2 + |e|2 (5.130) . H 0 H L2(Γ) 2 H1(Ω) EH ⊆Γ Using Young’s inequality, we get 1 0 ˜ B4 := H 2 k(K + 1)φH + WξH kL2(Γ) infkξ − ξH k 1 ˜ H 2 (Γ) ξH X 0 2 2 . Hk(K + 1)φH + WξH k 2 + kξ − ξH k 1 , (5.131) L (EH ) 2 H 2 (Γ) EH ⊆Γ and

1 ∂ 1 1 2 Ms ˜ B5 := H k (u0 − uH + VφH − (K + 1)ξH )kL2(Γ) infkφ − φH k − 1 ˜ H 2 (Γ) ∂s 2 2 φH

X ∂ Ms 1 1 2 2 . Hk (u0 − uH + VφH − (K + 1)ξH )k 2 + kφ − φH k − 1 ∂s 2 2 L (EH ) 2 H 2 (Γ) EH ⊆Γ (5.132)

2 2 2 We throw the terms |e| 1 , kξ − ξH k 1 and kφ − φH k 1 to the left hand side, H (Ω) H 2 (Γ) H− 2 (Γ) and we obtain the estimate of the theorem.

Theorem 5.7. (A posteriori error estimate for H > ) Assume that Lemma 5.2 and Theorem 5.2 hold for the transmission problem.

Ms Let (u, φ, ξ) be the solution of (5.100) and (uH , φH , ξH ) be the solution of the discrete problem (5.101). Then there holds the estimate:

1 2 Ms 2 2 2 kC ε(u − uH )kL (Ω)+kξ − ξH k 1 +kφ − φH k − 1 2 H 2 (Γ)/R H 2 (Γ) X Ms 2 Hkσ(u ) · n − t0 − φH k 2 . H L (EH ) EH ⊆Γ X ∂ Ms 1 1 2 + Hk (u0 − uH + VφH − (K + 1)ξH )k 2 ∂s 2 2 L (EH ) EH ⊆Γ X 0 2 + Hk(K + 1)φH + WξH k 2 L (EH ) EH ⊆Γ 1   2 X 2 X 2 X 2 X 2 X 2 + ηT + ηT + ηE + ηE +  kfk 2 ( ) h H h H L (TH ) H T ⊆Ω T ⊆Ω ˚ ˚ T ⊆Ω h H Eh⊆EH ⊆Ω EH ⊆Ω H

Proof. To prove the theorem, we ﬁrst denote

Ms 0 X 0 Ms uI (x) = IH u (x) = u (xj)Φj , (5.133) j

128 5.6 Coupling FEM-BEM

0 where u is the solution of the homogenized equation and uI is the interpolant of Ms u0,using the multiscale basis functions Φj , note that uI is diﬀerent from the deﬁnition Ms of IH u. We know that

0 L(uI ) = 0 in T, uI = IH u on ∂T (5.134)

From (5.115), we have

1 2 Ms 2 2 2 kC ε(u − uH )kL (Ω)+kξ − ξH k 1 +kφ − φhk − 1 2 H 2 (Γ)/R H 2 (Γ) Z Z Ms X Ms Ms . f(u − vH ) dx + [σ(uH ) · n]EH (u − vH ) ds Ω EH EH ⊆˚Ω Ms Ms − hσ(uH ) · n − t0 − φH , u − vH i 1 0 ˜ + H 2 k(K + 1)φH + WξH kL2(Γ) infkξ − ξH k 1 ˜ H 2 (Γ) ξH 1 ∂ 1 1 2 Ms ˜ + H k (u0 − uH + VφH − (K + 1)ξH )kL2(Γ) infkφ − φH k − 1 ˜ H 2 (Γ) ∂s 2 2 φH (5.135)

First, we estimate the term Z Ms B1 := f(u − vH ) dx (5.136) Ω

Ms We choose vH = uI , using (5.47) and Cauchy Schwarz inequality, we obtain X B ≤ kfk 2 ku − u k 2 1 L (TH ) I L (TH ) TH ⊆Ω X 2 1 ≤ kfk 2 (C1H kfk 2 +C2( ) 2 ) L (TH ) L (Ω) H TH ⊆Ω X 2 2 X 1 ≤ C1H kfk 2 + C2kfk 2 ( ) 2 L (TH ) L (TH ) H TH ⊆Ω TH ⊆Ω (5.137)

From (5.119), we obtain

1   2 X 2 X 2 X 2 B1 . ηT + ηT +  kfk 2 ( ) (5.138) h H L (TH ) H Th⊆Ω TH ⊆Ω TH ⊆Ω

Now we estimate the term Z X Ms Ms B2 := [σ(uH ) · n]EH (u − vH ) ds. (5.139) EH EH ⊆˚Ω

129 5 Extended Multiscale Finite Element Method in Linear Elasticity

Ms Using vH = IH u on ∂T , we obtain (see (5.122)-(5.128))

X 2 X 2 2 B2 η + η + |e| 1 . (5.140) . Eh EH 2 H (Ω) ˚ ˚ Eh⊆EH ⊆Ω EH ⊆Ω Similarly to Theorem 5.6, we obtain

Ms Ms B3 := hσ(uH ) · n − t0 − φH , u − vH i X Hkσ(uMs) · n − t − φ k2 + |e|2 (5.141) . H 0 H L2(Γ) 2 H1(Ω) EH ⊆Γ

1 0 ˜ B4 := H 2 k(K + 1)φH + WξH kL2(Γ) infkξ − ξH k 1 ˜ H 2 (Γ) ξH X 0 2 2 . Hk(K + 1)φH + WξH k 2 + kξ − ξH k 1 , (5.142) L (EH ) 2 H 2 (Γ) EH ⊆Γ and

1 ∂ 1 1 2 Ms ˜ B5 := H k (u0 − uH + VφH − (K + 1)ξH )kL2(Γ) infkφ − φH k − 1 ˜ H 2 (Γ) ∂s 2 2 φH

X ∂ Ms 1 1 2 2 . Hk (u0 − uH + VφH − (K + 1)ξH )k 2 + kφ − φH k − 1 ∂s 2 2 L (EH ) 2 H 2 (Γ) EH ⊆Γ (5.143)

2 2 2 We throw the terms |e| 1 , kξ − ξH k 1 and kφ − φH k 1 to the left hand side, H (Ω) H 2 (Γ) H− 2 (Γ) the estimate of the Theorem follows immediatly.

5.7 Numerical Experiments

Example 1: In this example we show the convergence of the EMsFEM for a homogeneous problem. The results obtained by the traditional FEM can be regarded as reference values. We choose Young’s modulus E = 2000 and Poisson’s ratio ν = 0.3.

The number of coarse elements is Mc × Nc, where Mc and Nc denote the numbers of the elements in the x and y directions, respectively. We consider a number of coarse elements Mc × Nc between 12 × 2 and 96 × 16. There are Mf × Nf fine scale elements within each coarse element. In Figure 5.4-5.8 we show the distributions of the mi- croscopic von-Mises stress obtained by the extended multiscale finite element method EMsFEM and the traditional finite element method FEM.

The diﬀerence between the EMsFEM and the traditional FEM can be observed for

Mc × Nc = 12 × 2. The results obtained by EMsFEM converge to the reference values as the number of elements Mc × Nc increases as shown in Figure 5.4-5.7

Example 2. In this example we assume that the Poisson’s ratio is a constant, ν = 0.3, and Young’s modulus varies in the range of 2.0 − 3.0 × 103 according to a uniform

130 5.7 Numerical Experiments distribution. Young’s modulus and the Poisson ratio are given as shown in Figure 5.9.

The length and the width of the model are given by Lx, Ly = L respectively where Lx = NxLy and Nx ∈ N. Here we choose L = 1 and Nx = 6. The small scale of the heterogeneity is denoted by Lh, and we assume that Young’s modulus is constant in each element Lh × Lh. The number of coarse elements is Mc × Nc, where Mc and Nc denote the numbers of coarse elements in the x and y directions, respectively. In Figure 5.10 we show the deformed conﬁguration for H = 0.062500. In this example we use a discretization of the size of the heterogeneity, Lh = h. For the FEM, good results are obtained only if the mesh size is smaller than the size of the heterogeneities Lh. The results obtained by the FEM can be regarded as reference values for Lh = 8h. The results obtained by the traditional FEM have relative large errors if Lh ≤ h [54]. The EMsFEM has higher accuracy than the traditional FEM. The numerically constructed basis functions capture the micro scale heterogeneities within each coarse element as shown in Figure 5.11.

131 5 Extended Multiscale Finite Element Method in Linear Elasticity

Figure 5.4: The distributions of the Mises stress obtained by the EMsFEM with

Mc × Nc = 12 × 2, Mf × Nf = 16 × 16, H = 0.5, h = 0.031250

Figure 5.5: The distributions of the Mises stress obtained by the EMsFEM with

Mc × Nc = 24 × 4, Mf × Nf = 8 × 8, H = 0.25, h = 0.031250

Figure 5.6: The distributions of the Mises stress obtained by the EMsFEM with

Mc × Nc = 48 × 8, Mf × Nf = 4 × 4, H = 0.125, h = 0.031250

Figure 5.7: The distributions of the Mises stress obtained by the EMsFEM with

Mc × Nc = 96 × 16, Mf × Nf = 2 × 2, H = 0.062500, h = 0.031250

Figure 5.8: The distributions of the Mises stress obtained by FEM with h = 0.031250

132 5.7 Numerical Experiments

Figure 5.9: The Young’s modulus and the Poisson’s ratio for the heterogeneous model

Figure 5.10: Deformed geometry

Figure 5.11: The distributions of the Mises stress obtained by the EMsFEM with

Mc × Nc = 96 × 16, H = 0.062500, Lh = h = 0.031250

Figure 5.12: The distributions of the Mises stress obtained by FEM with Lh = 0.031250 and h = 0.0039 (Lh = 8h)

133

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Curriculum Vitae

Personal Data: Name Abderrahman Issaoui Born 03.02.1977 in Metlaoui, Tunisia Citizenship Tunisian, German Family Status Married, one child Education and Employment:

1996 Baccalaureat (Abitur) at secondary school, Metlaoui 10/1996 – 05/2001 Student of Mathematics and Physics Faculte des sciences de Bizert (Tunisia) 05/2001 Maitrise in Mathematics 2001 – 2003 Teacher at a private school 2003–2005 German language courses 10/2005 – 03/2010 Start of studies of Mathematics with minor subject Physics at the Leibniz UniversitätHannover 03/2010 Diploma in Mathematics, topic of the Diploma thesis: Mathematische Analysis des Skin Effekts since 04/2010 Research assistant and Ph.D. student at the Institute for Applied Mathematics (IfAM) / IRTG 1627 at the Leibniz UniversitätHannover

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