A Feti-Dp Preconditioner for a Composite Finite Element and Discontinuous Galerkin Method

A FETI-DP PRECONDITIONER FOR A COMPOSITE FINITE ELEMENT AND DISCONTINUOUS GALERKIN METHOD

MAKSYMILIAN DRYJA∗, JUAN GALVIS †, AND MARCUS SARKIS ‡

Abstract. In this paper a Nitsche-type discretization based on discontinuous Galerkin (DG) method for an elliptic two-dimensional problem with discontinuous coefficients is considered. The problem is posed on a polygonal region Ω which is a union of N disjoint polygonal subdomains Ωi of diameter O(Hi). The discontinuities of the coefficients, possibly very large, are assumed to occur only across the subdomain interfaces ∂Ωi. Inside each subdomain, a conforming finite element space on a quasiuniform triangulation with mesh size O(hi) is introduced. To handle the nonmatching meshes across the subdomain interfaces, a discontinuous Galerkin discretization is applied only on the interfaces. For solving the resulting discrete system, a FETI-DP type method is proposed and analyzed. It is established that the condition number of the preconditioned linear 2 system is estimated by C(1 + maxi log Hi/hi) with a constant C independent of hi, hi/hj , Hi and the jumps of coefficients. The method is well suited for parallel computations and it can be extended to three-dimensional problems. Numerical results are presented to validade the theory.

Key words. Interior penalty discretization, discontinuous Galerkin, elliptic problems with discontinuous coeﬃcients, ﬁnite element method, FETI-DP algorithms, preconditioners

AMS subject classiﬁcations. 65F10, 65N20, 65N30

1. Introduction. In this paper we consider an elliptic problem with highly discontinuous coefficients. The problem is posed on a polygonal region Ω which is a union of disjoint 2-D polygonal subdomains Ωi of diameter O(Hi) and we assume N that this partition {Ωi}i=1 is geometrically conforming, i.e., ∀i 6= j the intersection ∂Ωi ∩∂Ωj is empty or is a common corner or edge of Ωi and Ωj . The discontinuities of the coefficients are assumed to occur only across ∂Ωi. The problem is approximated by a conforming finite element method (FEM) inside each Ωi, with hi as a mesh parameter, and nonmatching meshes are allowed to occur across the interfaces ∂Ωi. In order to deal with the nonconformity of the FE spaces across ∂Ωi, a discrete problem is formulated using the symmetric interior penalty DG method on the ∂Ωi only, see [33, 31]. This kind of composite discretization is motivated by the local regularity of the solution of the problem. This approach, which we denote by composite FE/DG method, falls in the class of Nitsche-type mortaring methods, see [4, 33, 20], where conforming FEMs can be explored inside the subdomains Ωi rather than using the full DG method everywhere, see [4, 11]. As a result, the composite FE/DG method approach will imply in essential computational savings both in terms of memory and CPU time. The main goal of this paper is to design and analyze a FETI-DP preconditioner for the resulting discrete system. To the best of our knowledge, the FETI-DP method has never been considered before in the literature for DG or composite FE/DG discretizations.

The ﬁrst FETI-DP method for standard continuous ﬁnite element discretization

∗Department of Mathematics, Warsaw University, Banacha 2, 00-097 Warsaw, Poland. This research was supported in part by the Polish Sciences Foundation under grant 2011/01/B/ST1/01179. ([email protected]). †Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA. ([email protected]). ‡Instituto Nacional de Matem´atica Pura e Aplicada (IMPA), Estrada Dona Castorina 110, CEP 22460-320, Rio de Janeiro, Brazil, and Department of Mathematical Sciences at Worcester Polytech- nic Institute, 100 Institute Road, Worcester, MA 01609, USA. ([email protected]). 1 2 Dryja, Galvis and Sarkis was introduced in [15] and is a nonoverlapping domain decomposition method that enforces continuity of the solution at subdomain interfaces by Lagrange multipliers except at subdomain corners where the continuity is enforced directly by assigning a unique value for the functions at each corner. The ﬁrst mathematical analysis of the method was provided in [30]. The method was further improved by enforcing the continuity directly on averages across the edges or faces on subdomain interfaces [16, 24, 34], resulting in better parallel scalability for three-dimensional problems. FETI-DP methods, and also methods of this class such as FETI, BDDC and BDD, have been tested very successfully and analyzed theoretically for a wide variety of problems and standard continuous discretizations, see [34] and references therein.

The main goal of this paper is to develop the FETI-DP methodology for a non- trivial problem using a composite FE/DG method. More specifically, we consider a scalar second order elliptic problem with discontinuous coefficients with nonmatching meshes across the subdomain interfaces. The discontinuities of the coefficients do not necessarily satisfy the quasi-monotonicity condition on the jumps of the coefficients, see [14]. In addition, the mesh size ratio hi/hj across the interfaces between two neigh- borings subdomains Ωi and Ωj are not necessary of O(1). In [11], two discretizations were introduced which are based on symmetric IPDG discretizations on the subdomain interfaces. These discretizations aim at solving problems in the presence of both nonmatching meshes and discontinuous coefficients across these interfaces. The first discretization is based on the fluxes that we get by integration by parts, while the second one the fluxes are modified using harmonic average of the coefficients. It turns out that the second discretization is easier for the preconditioning point of view since the nonsymmetric term of the bilinear form is smaller, hence it requires a smaller penalty term, and as a consequence, the latter term can be easily controled by the energy term of the bilinear form by using a proper Poincarré-Friedrich in- equalty. To the best of our knowledge, there is no literature on preconditioning for the first discretization, and this paper will fill this scientific gap. Furthermore, even for the second discretization, numerical results show, see [12, 13], that preconditioners based on Neumann-Neumann or BDDC with edge constraints require an interface mesh-coefficient condition to achieve robustness with respect to both jumps of coefficients and mesh sizes ratio across the interfaces. This condition is not needed for the preconditioner that we consider here in this paper. We expect that the methodology developed here can be extended to a large class of problems and for full DG discretizations, see [4, 21, 31]. In order to discuss why the development of FETI- DP preconditioners for composite FE/DG methods is more complicated compared to FETI-DP for conforming or mortar FE methods, we next give a brief introduction on FETI-DP for composite FE/DG method. A fully discussion and analysis of this method will be given in the remaining sections of this paper.

In this paper, the composite FE/DG discrete problem is reduced to the Schur complement problem with respect to unknowns on the interfaces of the subdomains Ωi. For that, discrete harmonic functions defined in a special way, i.e., in the DG sense, are used. We note that there are unknowns on both sides of the interfaces ∂Ωi, and on the contrary of the mortar methods, see [7], unknowns from one side of a interface cannot be eliminated via master-to-slave projections. This means that the unknowns on both sides of the interfaces should be kept as degrees of freedom of the linear system to be solved. Additionally, due to the DG penalty terms on the FETI-DP for a composite FE/DG discretization 3 interfaces ∂Ωi, these degrees of freedom are coupled across these interfaces. These issues characterize some of the main difficulties on designing and analyzing FETI-DP type methods for composite FE/DG discretizations. Distinctively from the classical conforming FE discretizations or from the mortar finite element discretizations, here a double layer of Lagrange multipliers are needed on interfaces rather than a single layer of Lagrange multipliers as normally is seen in FETI-DP for conforming or mortar FEMs. Despite the fact we follow the FETI-DP abstract approach from [34, 30], which was aimed to single layer of Lagrange multipliers, in this paper we successfully overcome this difficulty. The algorithm we develop in this paper is as follows. Let ′ ¯ ¯ ¯ Γi be the union of all edges Fij and Fji which are common to Ωi and Ωj , where Fij and F¯ji refer to the Ωi and Ωj sides, respectively, see Figure 3.1. We note that each ′ Γi has unknowns (degrees of freedom) corresponding to nodal points on the closure ¯ ′ ′ ∂Ωi\∂Ω and on the Fji ⊂ ∂Ωj . We now need to couple Γi with the other Γj . We ′ first impose continuity at the corners of Γi (which are corners of Ωi and common ¯ ′ endpoints of Fji) with the corners of the Γj , see Figure 4.1. These unknowns are ′ ′ called primal. The remaining unknowns on Γi and Γj are called dual and have jumps, hence, Lagrange multipliers are introduced to eliminate these jumps, see Figure 3.2. For the dual system with Lagrange multipliers, an special block diagonal precondi- ′ tioner is designed. It leads to independent local problems on Γi for 1 ≤ i ≤ N. It is proved that the proposed method is almost optimal with a condition number esti- 2 mate bounded by C(1 + maxi log Hi/hi) , where C does not depend on hi, hj , hi/hj , the number of subdomains Ωi and the jumps in the coefficients. The method can be extended to composite FE/DG discretizations of three-dimensional problems, see Section 7. The FETI-DP method developed here complements the BDDC methodology for the composite FE/DG discretizations developed recently in [12]. We note that the discretization used there (based on the harmonic average of the coefficients) is not the same as the one we consider here, additionally, the constraints used there are based on edges while here are based on corners. We point out that the introduction of corner constraints eliminates the interface condition assumption required in [12]. We note that other types of preconditioners have been considered for solving DG discretizations. In connection with block diagonal or overlapping Schwarz methods see for example [17, 18, 27, 8, 1, 2, 10, 29, 5] while for multilevel preconditioners [19, 22, 28, 26, 25, 6]. We note that these latter preconditioners do not use discrete harmonic extensions and does not analyze the cases of nonmatching meshes across subdomains and discontinuous coefficients.

The paper is organized as follows. In Section 2 the differential problem and a composite FE/DG discretization are formulated. In Section 3, the Schur complement problem is derived using discrete harmonic functions defined in a special way (in the DG sense). In Section 4, the so-called FETI-DP method is introduced, i.e., the Schur complement problem is reformulated by imposing continuity for the primal variables and by using Lagrange multipliers at the dual variables, and an special block diagonal preconditioner is defined. The main results of the paper are Theorem 4.3 and Lemma 4.5. Section 5 is devoted to the implementation of the FETI-DP method while in Sec- tion 6 numerical tests are reported which confirm the theoretical results. In Section 7, conclusions and extensions of the proposed preconditioner are discussed.

2. Diﬀerential and discrete problems. In this section we discuss the continuous and discrete problems we take into consideration for preconditioning. 4 Dryja, Galvis and Sarkis

∗ 1 2.1. Diﬀerential problem. Consider the following problem: Find uex ∈ H0 (Ω) such that

∗ 1 (2.1) a(uex, v)= f(v) ∀v ∈ H0 (Ω) where

N a(u, v) := ρi(x)∇u · ∇v dx and f(v) := fv dx. Z Z Xi=1 Ωi Ω

N We assume that Ω = ∪i=1Ωi and the substructures Ωi are disjoint shaped reg- N ular polygonal subregions of diameter O(Hi). We assume the partition {Ωi}i=1 is geometrically conforming, i.e., ∀i 6= j the intersection ∂Ωi ∩ ∂Ωj is empty or is a common corner or edge of Ωi and Ωj , where here and below an edge means a curve of continuous intervals and its two endpoints are called corners and the collection of these corners on ∂Ωi are referred to corners of Ωi. Let us denote F¯ij := ∂Ωi ∩ ∂Ωj as an edge of ∂Ωi and F¯ji := ∂Ωj ∩ ∂Ωi an edge of ∂Ωj . In spite of the common edge Fij and Fji being geometrically the same, they will be treated separately since we consider different triangulations on F¯ij ⊂ ∂Ωi with a mesh parameter hi and on F¯ji ⊂ ∂Ωj with a mesh parameter hj. Sometimes we use the notation Fijh and Fjih to refer the sets of nodal points of the triangulation on Fij and Fji with parameters hi and hj , respectively, and F¯ijh and F¯jih when the endpoints are included. For preconditioning analysis, the above notation will be more convenient for us than using the + − 2 usual notation Fij and Fij sometimes found in the literature. We assume f ∈ L (Ω), and for simplicity of presentation let ρi(x) be a positive constant ρi. 2.2. Discrete problem. Let us introduce a shape regular and quasiuniform triangulation with triangular elements in each Ωi and let hi be the mesh size. The resulting triangulation on Ω is in general nonmatching across ∂Ωi. Let Xi(Ωi) be the regular finite element (FE) space of piecewise linear and continuous functions in Ωi and define

N (2.2) X(Ω) = Xi(Ωi) ≡ X1(Ω1) × X2(Ω2) ×···× XN (ΩN ). iY=1

We note that we do not assume that functions in Xi(Ωi) vanish on ∂Ωi ∩ ∂Ω. 0 Let us denote by Ei the set of indices j of Ωj which has a common edge Fji with Ωi. To take into account also edges of Ωi which belong to ∂Ω, let us introduce a set ∂ of indices Ei to refer theses edges. The set of indices of all edges of Ωi is denoted 0 ∂ by Ei := Ei ∪Ei . A discrete problem obtained by a composite FE/DG method, see ∗ ∗ N [31, 4, 11], is of the form: Find u = {ui }i=1 ∈ X(Ω) where ui ∈ Xi(Ωi), such that

∗ N (2.3) ah(u , v)= f(v) ∀v = {vi}i=1 ∈ X(Ω), where

N N ′ (2.4) ah(u, v) := a (u, v) and f(v) := fvi dx, i Z Xi=1 Xi=1 Ωi

′ (2.5) ai(u, v) := ai(u, v)+ si(u, v)+ pi(u, v), FETI-DP for a composite FE/DG discretization 5 and

(2.6) ai(u, v) := ρi∇ui · ∇vi dx, ZΩi

1 ∂ui ∂vi (2.7) si(u, v) := ρi (vj − vi)+ ρi (uj − ui) ds ZFij lij ∂n ∂n jX∈Ei and

δ ρi (2.8) pi(u, v) := (uj − ui)(vj − vi) ds. ZFij li hij jX∈Ei

0 Here, when j ∈ Ei we set lij = 2 and let hij := 2hihj /(hi + hj ), i.e., the harmonic ∂ average of hi and hj . When j ∈Ei we denote the boundary edges Fij ⊂ ∂Ωi by Fi∂ and set li∂ = 1 and hi∂ = hi, and on the artiﬁcial edge Fji ≡ F∂i we set u∂ = 0 and ∂ v∂ = 0. The partial derivative ∂n denotes the outward normal derivative on ∂Ωi and δ is the penalty positive parameter.

We introduce the bilinear forms

(2.9) di(u, v) := ai(u, v)+ pi(u, v) and

N (2.10) dh(u, v) := di(u, v), Xi=1 and note that the norm deﬁned by dh(·, ·) is a broken norm in X(Ω) with weights given by ρ and δ ρi . For u = {u }N ∈ X(Ω) this discrete norm is deﬁned by i lij hij i i=1

N 2 2 δ ρi 2 2 k u kh:= dh(u,u)= ρi k ∇ui kL (Ωi) + (ui − uj ) ds . lij hij ZFij Xi=1  jX∈Ei    It is known that there exists a δ0 = O(1) > 0 and a positive constant c, which does not depend on ρi, Hi, hi and u, such that for every δ ≥ δ0, we obtain |si(u,u)|≤ cdi(u,u) and i |si(u,u)|≤ cdh(u,u), and therefore, the following lemma is valid: P Lemma 2.1. There exists δ0 > 0 such that for δ ≥ δ0 and for all u ∈ X(Ω) we have

′ (2.11) γ0di(u,u) ≤ ai(u,u) ≤ γ1di(u,u), 1 ≤ i ≤ N and

(2.12) γ0dh(u,u) ≤ ah(u,u) ≤ γ1dh(u,u).

Here, γ0 and γ1 are positive constants independent of the ρi,hi, Hi and u. For the proof we refer to [11, 12]. 6 Dryja, Galvis and Sarkis

Lemma 2.1 implies that the discrete problem (2.3) is elliptic and continuous, therefore, the solution exists and it is unique and stable. Lemma 2.1 together with Lemma 3.1, see below, are going to be fundamental for establishing condition number estimates for the FETI-DP preconditioned system developed in the remaining sections. An optimal a priori error estimate of this method was established in [3, 4] for the continuous coeﬃcient case. When the coeﬃcients are discontinuous across substructures and/or when hi and hj are not necessary of the same order, the following result is established in [11].

∗ ∗ ∗ Lemma 2.2. Let uex and u be the solution of (2.1) and (2.3). If uex|Ωi ∈ 1+s H (Ωi), 1 ≤ i ≤ N, and 1/2 ≤ s ≤ 1, then

N ∗ ∗ 2 2s hj 2s ∗ 2 1+s k uex − u kh≤ C hi + hj  ρikuexkH (Ωi) hi Xi=1 jX∈E0  i  ∗ where C is independent of hi, Hi, ρi and uex. We note that recently in [9], for matching meshes, it was established a quasi-optimal a priori error estimate whose solutions are only in H1+s smooth with s ∈ (0, 1/2], i.e., covering the case in which less regularity of the solutions is assumed. 3. Schur complement systems and discrete harmonic extensions. The ﬁrst step of many iterative substructuring solvers, such as the FETI-DP method that we consider in this paper, requires the elimination of unknowns interior to the subdomains. In this section, we describe this step for composite FE/DG discretizations.

′ Fig. 3.1. Illustration of the node classiﬁcation on Ωi.

We introduce, see Figure 3.1, some notation and then formulate (2.3) as a variational problem with constraints. Let us introduce

′ Ω := Ωi {∪ 0 F¯ji} i j∈Ei [ ¯ 0 i.e., the union of Ωi and the Fji ⊂ ∂Ωj such that j ∈Ei , and let ′ Γi := ∂Ωi\∂Ω and Γ := Γi {∪ 0 F¯ji}. i j∈Ei [ FETI-DP for a composite FE/DG discretization 7

We also introduce the sets

N N N ′ ′ ′ ′ (3.1) Γ := Γi, Γ := Γi, Ii := Ωi\Γi and I := Ii. i[=1 iY=1 iY=1

′ ′ Let Wi(Ωi) be the FE space of functions deﬁned by the nodal values on Ωi

′ ¯ (3.2) Wi(Ωi)= Wi(Ωi) × Wi(Fji), 0 jY∈Ei

¯ ¯ 0 where Wi(Fji) is the trace of the FE space Xj (Ωj ) on Fji ⊂ ∂Ωj for all j ∈ Ei . A ′ ′ ′ function ui ∈ Wi(Ωi) is deﬁned by the nodal values on Ωi, i.e., by the nodal values ¯ 0 ′ on Ωi and on Fji for all j ∈ Ei . Below, we will denote ui by ui if it is not confused ′ with functions of Xi(Ωi). A function ui ∈ Wi(Ωi) will be represented as

ui = {(ui)i, {(ui)j } 0 }, j∈Ei where (u ) := u (u restricted to Ω ) and (u ) := u ¯ (u restricted to F¯ ). i i i |Ωi i i i j i |Fji i ji Here and below we use the same notation to denote both FE functions and their ′ ′ ′ vectors representations. Note that ai(·, ·), see (2.5), is defined on Wi(Ωi) × Wi(Ωi) ′ with corresponding stiffness matrix Ai defined by

′ ′ ′ (3.3) ai(ui, vi)= hAiui, vii ui, vi ∈ Wi(Ωi), where hui, vii denotes the ℓ2 inner product associated to nodes associated to the vector ′ space in consideration. We also will represent ui ∈ Wi(Ωi) as ui = (ui,I ,ui,Γ′ ) where ′ ui,Γ′ represents values of ui at nodal points on Γi and ui,I at the interior nodal points ′ ′ on Ii, see (3.1). Hence, let us represent Wi(Ωi) as the vector spaces Wi(Ii) × Wi(Γi). ′ Using the representation ui = (ui,I ,ui,Γ′ ), the matrix Ai can be represented as

′ ′ ′ Ai,II Ai,IΓ′ (3.4) Ai = ′ ′ , Ai,Γ′I Ai,Γ′Γ′

′ where the block rows and columns correspond to the nodal points of Ii and Γi, respectively.

′ The Schur complement of Ai with respect to ui,Γ′ is of the form:

′ ′ ′ ′ −1 ′ (3.5) Si := Ai,Γ′Γ′ − Ai,Γ′I (Ai,II ) Ai,IΓ′ .

′ Note that Si satisﬁes

′ ′ (3.6) hSiui,Γ′ ,ui,Γ′ i = min ai(wi, wi),

′ where the minimum is taken over wi = (wi,I , wi,Γ′ ) ∈ Wi(Ωi) such that wi,Γ′ = ui,Γ′ ′ ′ ′ on Γi. The bilinear form ai(·, ·) is symmetric and nonnegative with respect to Wi(Ωi), see Lemma 2.1. The minimizing function satisfying (3.6) is called discrete harmonic ′ ′ in the sense of ai(·, ·) or in the sense of Hi. An equivalent deﬁnition of the minimizing ′ ′ function Hiui,Γ′ ∈ Wi(Ωi) is given by the solution of

o ′ ′ ′ (3.7) ai(Hiui,Γ′ , vi)=0 vi ∈ W i(Ωi), 8 Dryja, Galvis and Sarkis

′ ′ (3.8) Hiui,Γ′ = ui,Γ′ on Γi,

o ′ ′ ′ where W i(Ωi) is the subspace of Wi(Ωi) of functions which vanish on Γi. We note ′ that for substructures Ωi which intersect ∂Ω by edges, the nodal values of Wi(Ωi) on ′ ∂Ωi\Γi are treated as unknowns and belong to Ii.

′ Let us also introduce Hiui,Γ′ ∈ Wi(Ωi) as the standard discrete harmonic function ′ ′ of ui,Γ′ ∈ Wi(Γi), i.e., Hiui,Γ′ = ui,Γ′ on Γi and discrete harmonic in Ωi in the sense of ai(·, ·), i.e.,

o ′ ai(Hiui,Γ′ , vi)=0 vi ∈ W i(Ωi)

′ where we reinterpretate ai(·, ·), see (2.6), as a bilinear form with respect to Wi(Ωi). ′ We note that the extensions Hi and Hi diﬀer from each other since Hiui,Γ′ at the in- ′ terior nodes Ii depends only on the nodal values of ui,Γ′ on Γi, while Hiui,Γ′ depends ′ on the nodal values of ui,Γ′ on Γi. The following lemma shows the equivalence (in the energy form deﬁned by di(·, ·)) between discrete harmonic functions in the sense ′ of Hi and in the sense of Hi; for the proof see Lemma 4.1 of [12]. This equivalence allows us to take advantages of all the discrete Sobolev results known for Hi discrete harmonic extensions.

′ Lemma 3.1. For ui,Γ′ ∈ Wi(Γi) it holds that

′ ′ (3.9) di(Hiui,Γ′ , Hiui,Γ′ ) ≤ di(Hiui,Γ′ , Hiui,Γ′ ) ≤ Cdi(Hiui,Γ′ , Hiui,Γ′ ) where C is a positive constant independent of hi, Hi, ρi and ui,Γ′ .

The next corollary follows directly from Lemma 3.1 and Lemma 2.1.

′ Corollary 3.2. For ui,Γ′ ∈ Wi(Γi) it holds that

′ ′ ′ (3.10) C0di(Hiui,Γ′ , Hiui,Γ′ ) ≤ ai(Hiui,Γ′ , Hiui,Γ′ ) ≤ C1di(Hiui,Γ′ , Hiui,Γ′ ) where C0 and C1 are positive constants independent of hi, Hi, ρi and ui,Γ′ .

Let us introduce the product space

N ′ ′ (3.11) W (Ω ) := Wi(Ωi), iY=1

′ N ′ i.e., u ∈ W (Ω ) means that u = {ui}i=1 where ui ∈ Wi(Ωi); see (3.2) for the deﬁnition ′ of W (Ω ). We remind that (u ) = u (u restricted to Ω ) and (u ) = u ¯ (u i i i i i |Ωi i i i j i |Fji i restricted to F¯ji). Using the representation ui = (ui,I ,ui,Γ′ ) where ui,I ∈ Wi(Ii) and ′ ui,Γ′ ∈ Wi(Γi), see (3.4), let us introduce the product space

N ′ ′ (3.12) W (Γ ) := Wi(Γi), iY=1

′ N ′ ′ i.e., uΓ′ ∈ W (Γ ) means that uΓ′ = {ui,Γ′ }i=1 where ui,Γ′ ∈ Wi(Γi). The space W (Γ ) which was deﬁned on Γ′ only, also will be interpreted below as the subspace of W (Ω′) FETI-DP for a composite FE/DG discretization 9

′ of functions which are discrete harmonic in the sense of Hi in each Ωi.

We now consider the subspace Wˆ (Ω′) ⊂ W (Ω′) and Wˆ (Γ′) ⊂ W (Γ′) as the space of functions which are continuous on Γ in the sense of the following deﬁnition (for notation see (3.1)):

Fig. 3.2. Illustration of the continuity on Γ.

ˆ ′ ˆ ′ N ′ Definition 3.3. (Spaces W (Ω ) and W (Γ )). We say that u = {ui}i=1 ∈ W (Ω ) is continuous on Γ if for all 1 ≤ i ≤ N we have ¯ 0 (3.13) (ui)i(x) = (uj )i(x) for all x ∈ Fij for all j ∈ Ei and ¯ 0 (3.14) (ui)j (x) = (uj )j (x) for all x ∈ Fji for all j ∈ Ei . In Figure 3.2 we illustrate this continuity by assigning the same nodal value for nodes connected by a line. The subspace of W (Ω′) of continuous functions on Γ is denoted by Wˆ (Ω′), and the subspace of Wˆ (Ω′) of functions which are discrete harmonic in the ′ ˆ ′ sense of Hi in each Ωi is denoted by W (Γ ).

Note that there is a one-to-one correspondence between vectors in the spaces X(Ω) ′ ′ and Wˆ (Ω ). Indeed, let us introduce the restriction matrices R ′ : X(Ω) → W (Ω ) Ωi i i N which assign uniquely the vector values of u = {ui}i=1 ∈ X(Ω) where ui ∈ Xi(Ωi), see ′ ¯ 0 (2.2), to vi ∈ Wi(Ωi) deﬁned by (vi)i = ui on Ωi and (vi)j = uj on Fji for all j ∈Ei . N It is easy to see that v := {v } where v := R ′ u belongs to Wˆ (Ω). And vice- i i=1 i Ωi N ˆ N versa, for each v = {vi}i=1 ∈ W (Ω), we can deﬁne uniquely u = {ui}i=1 ∈ X(Ω) by setting ui = (vi)i. Since u and v have identical nodal values, we will refer sometimes u ∈ Wˆ (Ω′) or u ∈ X(Ω). For instance, the solution u∗ of (2.3) can be interpreted as a function in Wˆ (Ω′) or in X(Ω).

Note that the discrete problem (2.3) can be written as a system of algebraic equations (3.15) Auˆ ∗ = f 10 Dryja, Galvis and Sarkis

∗ N for u ∈ X(Ω) using the standard FE basis functions, and f = {fi}i=1 ∈ X(Ω), where fi is the load vector associated with individual subdomains Ωi, i.e., is Ω fvi when vi are the canonical basis functions of Xi(Ωi). The stiﬀness matrix Aˆ canR be obtained ′ ′ by assembling the matrices Ai, see (3.3), from W (Ω ) to X(Ω) as

N T ′ Aˆ = R ′ A RΩ′ . Ωi i i Xi=1 Note that the matrix Aˆ is no longer block diagonal since there are couplings between substructures due to the continuity on Γ, see Deﬁntion 3.3.

Note also that X(Ω) can be componentwise represented by X(I)×X(Γ), denoted N also by X(Ω), where X(I) := i=1 Xi(Ii) is the vector space of functions deﬁned by N nodal values on Ii, and X(Γ)Q := i=1 Xi(Γi) by the nodal values on Γi, see (3.1) and Figure 3.1. Hence, we can representQ u ∈ X(Ω) as u = (uI ,uΓ) with uI ∈ X(I) ′ and u ∈ X(Γ). We introduce the restriction R ′ : X(Γ) → W (Γ ) by assigning Γ Γi i i ′ ′ values uΓ ∈ X(Γ) into ui ∈ Wi(Γi) at the nodes of Γi. By eliminating the variable ∗ ∗ N ∗ ∗ ∗ uI = {ui,I }i=1 of u = (uI ,uΓ) from (3.15), see (3.4) and (3.5), it is easy to see that ˆ ∗ (3.16) SuΓ =ˆgΓ where

N N ˆ T ′ T ′ ′ −1 (3.17) S = R ′ S RΓ′ andg ˆΓ = fΓ − R ′ A ′ (A ) fi, Γi i i Γi i,Γ I i,II Xi=1 Xi=1

N where fΓ := {fi,Γ}i=1 with fi,Γ ∈ Xi(Γi), where the load vector fi,Γ is deﬁned by fvi,Γ when vi,Γ are the cannonical basis functions of Xi(Ωi) associated to nodes Ωi Ron Γi. It is also easy to see from (3.7) and (3.8) that

T ′ ′ ′ vi,I Ai,II Ai,IΓ′ Hiui,Γ′ ′ (3.18) ′ ′ = hSiui,Γ′ , vi,Γ′ i. vi,Γ′ Ai,Γ′I Ai,Γ′Γ′ . ui,Γ′

Note that Wˆ (Γ′) is the natural space for deﬁning hSˆ ·, ·i due to (3.17), (3.18) and the continuity of Wˆ (Γ′) on Γ, see Deﬁnition 3.3. 4. FETI-DP with corner constraints. We now design a FETI-DP method for solving (3.16). We follow the abstract approach described in pages 160-167 in [34].

For 1 ≤ i ≤ N, we introduce the nodal points associated to the corner unknowns, see Figure 3.1, by

′ (4.1) V := Vi {∪ 0 ∂Fji} where Vi := {∪ 0 ∂Fij }. i j∈Ei j∈Ei [ We now consider the subspace W˜ (Ω′) ⊂ W (Ω′) and W˜ (Γ′) ⊂ W (Γ′) as the space of functions which are continuous on all the Vi in the sense of the following deﬁnition:

˜ ′ ˜ ′ N Definition 4.1. (Subspaces W (Ω ) and W (Γ )). We say that u = {ui}i=1 ∈ ′ ′ W (Ω ) is continuous at the corners Vi if for 1 ≤ i ≤ N we have 0 (4.2) (ui)i(x) = (uj )i(x) at x ∈ ∂Fij for all j ∈ Ei FETI-DP for a composite FE/DG discretization 11

Fig. 4.1. Illustration of the continuity at the corners and

0 (4.3) (ui)j (x) = (uj )j (x) at x ∈ ∂Fji for all j ∈ Ei .

In Figure 4.1 we illustrate this continuity by assigning the same nodal value for nodes (corners) connected by a line. The subspace of W (Ω′) of continuous functions at the ′ ˜ ′ ˜ ′ corners Vi for all 1 ≤ i ≤ N is denoted by W (Ω ), and the subspace of W (Ω ) of ′ ˜ ′ functions which are discrete harmonic in the sense of Hi is denoted by W (Γ ).

Note that

(4.4) Wˆ (Γ′) ⊂ W˜ (Γ′) ⊂ W (Γ′).

˜ ′ Let A be the stiﬀness matrix which is obtained by assembling the matrices Ai for 1 ≤ i ≤ N, from W (Ω′) to W˜ (Ω′). Note that the matrix A˜ is no longer block diagonal ′ since there are couplings between variables at the corners Vi for 1 ≤ i ≤ N. We will ′ represent u ∈ W˜ (Ω ) as u = (uI ,uΠ,u△) where the subscript I refers to the interior N ′ degrees of freedom at nodal points I = i=1 Ii, the Π refers to the corners Vi for all 1 ≤ i ≤ N, and the △ refers to the remainingQ nodal points, i.e., the nodal points ′ ′ ˜ ′ Γi\Vi, for all 1 ≤ i ≤ N. The vector u = (uI ,uΠ,u△) ∈ W (Ω ) is obtained from the N ′ vector u = {ui}i=1 ∈ W (Ω ) using the equations (4.2), (4.3), i.e., the continuity of u ′ ˜ ′ on Vi for all 1 ≤ i ≤ N. Using the decomposition u = (uI ,uΠ,u△) ∈ W (Ω ) we can partition A˜ as

′ ′ ′ AII AIΠ AI△ ˜ ′ ˜ ′ (4.5) A =  AΠI AΠΠ AΠ△  . A′ A′ A′  △I △Π △△  We note that the only couplings across subdomains are through the variables Π where the matrix A˜ is subassembled. 12 Dryja, Galvis and Sarkis

A Schur complement of A˜ with respect to the △-unknowns (eliminating the I- and the Π-unknowns) is of the form

′ ′ −1 ′ ˜ ′ ′ ′ AII AIΠ AI△ (4.6) S := A△△ − (A△I A△Π) ′ ˜ ′ . AΠI AΠΠ AΠ△

′ A vector u ∈ W˜ (Γ ) can uniquely be represented by u = (uΠ,u△), therefore, we ′ ′ ′ ′ can represent W˜ (Γ ) = Wˆ Π(Γ ) × W△(Γ ), where Wˆ Π(Γ ) refers to the Π-degrees of ′ ′ ′ freedom of W˜ (Γ ) while W△(Γ ) to the △-degrees of freedom of W˜ (Γ ). The vector ′ space W△(Γ ) can be decomposed as

N ′ ′ (4.7) W△(Γ )= Wi,△(Γi) iY=1 ′ where the local space Wi,△(Γi) refers to the degrees of freedom associated to the ′ ′ ˜ ′ nodes of Γi\Vi for i = 1 ≤ i ≤ N. Hence, a vector u ∈ W (Γ ) can be repre- ˆ ′ N ′ sented as u = (uΠ,u△) with uΠ ∈ WΠ(Γ ) and u△ = {ui,△}i=1 ∈ W△(Γ ) where ′ ˜ ′ ui,△ ∈ Wi,△(Γi). Note that S, see (4.6), is deﬁned on the vector space W△(Γ ), and the following lemma follows (cf. Lemma 6.22 in [34] and Lemma 4.2 in [30]):

′ Lemma 4.2. Let u△ ∈ W△(Γ ) and let S˜ and A˜, deﬁned in (4.6) and (4.5), respectively. Then,

(4.8) hSu˜ △,u△i = minhAw,w˜ i where the minimum is taken over w = (wI , wΠ, w△) ∈ W˜ (Ω) such that w△ = u△.

′ ′ ′ Let us take u ∈ W˜ (Γ ) as u = (uΠ,u△) with uΠ ∈ Wˆ Π(Γ ) and u△ ∈ W△(Γ ), N ′ ′ where u△ = {ui,△}i=1 with ui,△ ∈ Wi,△(Γi). The vector ui,△ ∈ Wi,△(Γi) can be partitioned as

ui,△ = {(ui,△)i, {(ui,△)j } 0 } j∈Ei where

(ui,△)i = u and (ui,△)j = u . i,△|Γi\Vi i,△|Fji

′ In order to measure the jump of u△ ∈ W△(Γ ) across the △-nodes let us introduce the space Wˆ △(Γ) deﬁned by

N Wˆ △(Γ) = Xi(Γi\Vi), iY=1 where Xi(Γi\Vi) is the restriction of Xi(Ωi) to Γi\Vi, see Figure 3.1. To deﬁne the ′ ˆ N ′ jumping matrix B△ : W△(Γ ) → W△(Γ), let u△ = {ui,△}i=1 ∈ W△(Γ ) and let N ˆ v := B△u where v = {vi}i=1 ∈ W△(Γ) is deﬁned by 0 (4.9) vi = (ui,△)i − (uj,△)i on Fijh for all j ∈Ei .

The jumping matrix B△ can be written as

(1) (2) (N) (4.10) B△ = (B△ ,B△ , · · · ,B△ ), FETI-DP for a composite FE/DG discretization 13

(i) where the rectangular matrix B△ consists of columns of B△ attributed to the (i) ′ ′ components of functions of Wi,△(Γi) of the product space W△(Γ ), see (4.7). The entries of the rectangular matrix consist of values of {0, 1, −1}. It is easy to see that ′ the Range B△ = Wˆ △(Γ), B△ is full rank. In addition, if u = (uΠ,u△) ∈ W˜ (Γ ) and ′ B△u△ = 0 then u ∈ Wˆ (Γ ).

We can reformulate the problem (3.16) as the variational problem with constraints ′ ∗ ′ in W△(Γ ) space: Find u△ ∈ W△(Γ ) such that

∗ (4.11) J(u△) = min J(v△)

′ subject to v△ ∈ W△(Γ ) with constraints B△v△ = 0, where

(4.12) J(v△):=1/2hSv˜ △, v△i − hg˜△, v△i where S˜ is deﬁned in (4.6) and

′ ′ −1 ′ ′ AII AIΠ fI (4.13) g˜△ := f△ − (A△I A△Π) ′ ˜ . AΠI AΠΠ fΠ

N We note that f = {fi}i=1 ∈ X(Ω) was defined in (3.15) and it can be represented as ′ f = (fI ,fΠ,fΓ\Π). It remains to define f△ in (4.13). The forcing term f△ ∈ W△(Γ ) N is defined by f△ = {fi,△}i=1 where fi,△ are the load vectors associated with the individual subdomains Ωi, i.e., the entries fi,∆ are defined as fvi,∆dx when vi,∆ Ωi ′ ˜ R are the canonical basis functions of Wi,∆(Γi). Note that S is symmetric and positive definite since A˜ has these properties; see also Lemma 4.2. Introducing Lagrange multipliers λ ∈ Wˆ △(Γ), the problem (4.11) reduces to the saddle point problem of the ∗ ′ ∗ ˆ form: Find u△ ∈ W△(Γ ) and λ ∈ W△(Γ) such that

˜ ∗ T ∗ Su△ + B△λ =g ˜△ (4.14) ∗ B△u△ = 0.

Hence, (4.14) reduces to

(4.15) F λ∗ = g where

˜−1 T ˜−1 (4.16) F := B△S B△, g := B△S g˜△.

∗ ∗ When λ is computed, u△ can be found by solving the problem

˜ ∗ T ∗ (4.17) Su△ =g ˜△ − B△λ .

4.1. Dirichlet Preconditioner. We now deﬁne the FETI-DP preconditioner ′ ′ for F , see (4.16). Let Si,△ be the Schur complement of Si, see (3.5), restricted to ′ ′ ′ ′ ′ Wi,△(Γi) ⊂ Wi(Γi), i.e., taken Si on functions in Wi(Γi) which vanish on Vi. Let

′ ′ N (4.18) S△ := diag{Si,△}i=1. 14 Dryja, Galvis and Sarkis

′ ′ In other words, Si,△ is obtained from Si by deleting rows and columns corresponding ′ ′ to nodal values at nodal points of Vi ⊂ Γi.

(i) ′ ′ Let us introduce diagonal scaling matrices D△ : Wi,△(Γi) → Wi,△(Γi), for 1 ≤ (i) ′ ′ i ≤ N. The diagonal entry of D△ associated to a node x ∈ Γi\Vi, which we denote (i) by D△ (x), is deﬁned by

β (i) ρj 0 (4.19) D△ (x)= β β for x ∈{Fijh ∪ Fjih} for j ∈Ei , ρi + ρj for β ∈ [1/2, ∞), see [32], and deﬁne

(1) (1) (N) (N) (4.20) BD,△ = (B△ D△ , · · · ,B△ D△ ).

Let

T (4.21) P△ := BD,△B△

′ N ′ which maps W△(Γ ) into itself. It is easy to check that for w△ = {wi,△}i=1 ∈ W△(Γ ) and v△ := P△w△, the following equalities hold:

β ρj (4.22) (vi,△)i(x)= β β [(wi,△)i(x) − (wj,△)i(x)], x ∈ Fijh, ρi + ρj

β ρj (4.23) (vi,△)j (x)= β β [(wi,△)j (x) − (wj,△)j (x)], x ∈ Fjih. ρi + ρj

Note that

β ρi (4.24) (vj,△)j (x)= β β [(wj,△)j (x) − (wi,△)j (x)], x ∈ Fjih, ρi + ρj

β ρi (4.25) (vj,△)i(x)= β β [(wj,△)i(x)) − (wi,△)i(x))], x ∈ Fijh. ρi + ρj

By subtracting (4.25) from (4.22) and (4.23) from (4.24) we see that P△ preserves jumps in the sense that

(4.26) B△P△ = B△.

2 From this follows that P△ is a projection (P△ = P△).

In the FETI-DP method, the preconditioner M −1 is deﬁned as follows:

N −1 ′ T (i) (i) ′ (i) (i) T (4.27) M = BD,△S△BD,△ = B△ D△ Si,△D△ (B△ ) . Xi=1 FETI-DP for a composite FE/DG discretization 15

−1 ′ Note that M is a block-diagonal matrix, and each block is invertible since Si,△ and (i) (i) D△ are invertible and B△ is a full rank matrix. The following theorem holds:

Theorem 4.3. For any λ ∈ Wˆ △(Γ) it holds that

H (4.28) hMλ, λi ≤ hF λ, λi ≤ C(1 + log )2hMλ, λi h where C is a positive constant independent of hi, hi/hj, Hi, λ and the jumps of ρi. Here and below, log( H ) := maxN log( Hi ). h i=1 hi

Proof. We follow to the general abstract theory for FETI-DP methods developed in Theorem 6.35 of [34]. This abstract theory relies only on duality and linear algebra arguments, and properties such as that B△ is full rank, P△ is a projection, S˜ is invertible and the subspace inclusion (4.4). Using the same abstract arguments, the proof of the theorem follows by checking the Lemma 4.4 and Lemma 4.5, see below. The proofs of these two lemmas are not algebraic and they are dependent of the problem. The proofs of these two lemmas are presented separately below.

′ Lemma 4.4. For u△ ∈ W△(Γ ) it follows that

˜ ′ (4.29) hSu△,u△i ≤ hS△u△,u△i.

Proof. The proof follows from Lemma 4.2 and from

˜ ˜ ˜ ′ (4.30) hSu△,u△i = minhAw,wi ≤ minhAv,vi = hS△u△,u△i

′ where the minima are taken over w = (wI , wΠ, w△) ∈ W˜ (Ω ) such that w△ = u△, ′ and v = (vI , vΠ, v△) ∈ W˜ (Ω ) such that vΠ = 0 and v△ = u△.

′ Lemma 4.5. For any u△ ∈ W△(Γ ) it holds that

2 H 2 2 (4.31) k P△u△ kS′ ≤ C(1 + log ) k u△ k ˜ △ h S where C is a positive constant independent of hi, hi/hj, Hi, u△ and the jumps of ρi. Proof. We ﬁrst consider the case when the edges Fij are a single interval only. ′ ′ Let u△ ∈ W△(Γ ) and let u = (uΠ,u△) ∈ W˜ (Γ ) be the solution of

′ ′ (4.32) hSu˜ △,u△i = minhS w, wi =: hS u,ui,

′ ′ where the minimum is taken over w = (wΠ, w△) ∈ W˜ (Γ ) such that wΠ ∈ Wˆ Π(Γ ) and w△ = u△. Hence, we can replace ku△kS˜ in (4.31) by kukS′ .

N ′ Let us represent the u defined above as {ui}i=1 ∈ W (Γ ) where ui ∈ Wi(Γi). Let ¯ IFij (ui)i be the linear function on Fij defined by the values of (ui)i at x ∈ ∂Fij , and ¯ let IFji (ui)j be the linear function on Fji defined by the values of (ui)j at x ∈ ∂Fji. N ′ Letu ˆ := {uî}i=1 whereu î ∈ Wi(Γi) is defined by ¯ 0 (ûi)i = IFij (ui)i on Fijh for all j ∈ Ei 16 Dryja, Galvis and Sarkis and

¯ 0 (ˆui)j = IFji (ui)j on Fjih for all j ∈ Ei .

′ Note thatu ˆ ∈ Wˆ (Γ ), therefore, let us representu ˆ = (ˆuΠ, uˆ△) where B△uˆ△ = 0. Using this we have, see (4.21),

T T P△u△ ≡ BD,△B△u△ = BD,△B△(u△ − uˆ△)= P△(u△ − uˆ△).

Note that u − uˆ = 0 at the Π-nodes, hence, let us deﬁne v ∈ W (Γ′) to be equal to P△(u△ − uˆ△) at the △-nodes and equal to zero at the Π-nodes. Let us represent N ′ v = {vi}i=1 where vi ∈ Wi(Γi). We have

N 2 2 2 (4.33) k P△u△ k ′ =k v k ′ = k vi k ′ S∆ S Si Xi=1

′ ′ in view of the deﬁnition of Si,△ and S∆, see (4.18), (3.5) and (4.6), hence, to prove the lemma it remains to show that

N 2 2 2 (4.34) k vi k ′ ≤ C(1 + log H/h) kuk ′ Si S Xi=1 since by (4.32) we obtain (4.31). By Corollary 3.2 we need to show

N N 2 (4.35) d˜i(vi, vi) ≤ C(1 + log H/h) d˜i(ui,ui) Xi=1 Xi=1 where, see (2.9),

d˜i(vi, vi) := di(Hivi, Hivi) and so

2 ρiδ 2 ˜ 2 2 (4.36) di(vi, vi)= ρi k ∇(vi)i kL (Ωi) + k (vi)i − (vi)j kL (Fij ), lij hij jX∈Ei where (vi)i = Hivi and (ui)i = Hiui inside of the subdomains Ωi.

We ﬁrst estimate the ﬁrst term of (4.36). We have

2 2 2 1 2 (4.37) k ∇(vi)i kL (Ωi)≤ C k (vi)i k / H00 (Fij ) 0 jX∈Ei by the well-known estimate, see [34], and the fact that (vi)i = 0 at corners of ∂Ωi. Note that (4.37) is also valid for subdomains Ωi which intersect ∂Ω by edges since we use the obvious inequality

2 2 2 2 2 1/2 k ∇(vi)i kL (Ωi)≤k ∇(˜vi)i kL (Ωi)≤ C k (vi)i k H00 (Fij ) 0 jX∈Ei FETI-DP for a composite FE/DG discretization 17 where (˜vi)i is the standard discrete harmonic extension on Ωi with (˜vi)i = (vi)i on ¯ 0 ¯ ∂ edges Fij for j ∈Ei , and (˜vi)i = 0 on edges Fij for j ∈Ei . For the case Fij such that 0 j ∈Ei , we use (4.22) to get 2β 2 ρiρj 2 ρi k (vi)i k 1/2 = k (ui − uˆi)i − (uj − uˆj )i k 1/2 ≤ H00 (Fij ) β β 2 H00 (Fij ) (ρi + ρj ) 2 2 (4.38) ≤ 3 {ρi k (ui − uˆi)i k 1/2 +ρj k (uj − uˆj )j k 1/2 + H00 (Fij ) H00 (Fji ) 2β ρiρj 2 + k (uj − uˆj)i − (uj − uˆj )j k 1/2 }, β β 2 H00 (Fij ) (ρi + ρj )

2β ρiρj where we have used that β β 2 ≤ min{ρi,ρj } if β ∈ [1/2, ∞), see [32]. (ρi +ρj ) For estimating the ﬁrst term of the right-hand side of (4.38), it is well-known that, see [34],

2 Hi 2 2 1 2 2 (4.39) ρi k (ui − uˆi)i k / ≤ C(1 + log ) ρi k ∇(ui)i kL (Ωi)≤ H00 (Fij ) hi Hi 2 ≤ C(1 + log ) di(ui,ui) hi since (ˆui)i = IFij (ui)i is the linear interpolant of (ui)i on Fij with nodal values (ui)i given on ∂Fij . The estimate for the second term of the right-hand side of (4.38) is similar.

It remains to estimate the third term of the right-hand side of (4.38). We have 2 (4.40) k (uj − uˆj)i − (uj − uˆj)j k 1/2 = H00 (Fij ) 2 2 ((uj − uˆj )i − (uj − uˆj)j ) = |(u − uˆ ) − (u − uˆ ) | 1/2 + ds. j j i j j j H (Fij ) ZFij dist(s,∂Fij )

The first term of (4.40) is estimated as follows. Let Qi be the L2 projection on Xi(Fij ), the restriction of Xi(∂Ωi) to F¯ij with hi -triangulation on Fij . Using the 1/2 inverse inequality, the H and L2 stabilities of the L2 projection we have 2 2 (4.41)|(u − uˆ ) − (u − uˆ ) | 1/2 ≤ C {|Q ((u ) − (u ) )| 1/2 + j j i j j j H (Fij ) i j i j j H (Fij ) 2 2 2 +|Q (u − uˆ ) )| 1/2 + |(u − uˆ ) | 1/2 + |(û ) − (û ) | 1/2 } i j j j H (Fij ) j j j H (Fij ) j i j j H (Fij )

1 2 2 2 ≤ C{ k (u ) − (u ) k 2 +|(û ) − (û ) | 1/2 + |(u − uˆ ) | 1/2 }≤ j i j j L (Fij ) j i j j H (Fij ) j j j H (Fij ) hi 1 2 2 Hj 2 2 2 2 ≤ C{ k (uj )i − (uj)j kL (Fij ) +max((uj )i − (uj )j ) +(1+log ) k ∇(uj )j kL (Ωj )} hi ∂Fij hj 0 since (ûj )i and (ûj )j are linear on Fij and Fji, respectively, for j ∈ Ei . The second term of right-hand side of last inequality of (4.41) is estimated as follows. Let (¯uj )j be the average of (uj )j on Fji. We obtain 2 2 max((uj )i − (uj )j ) ≤ 3 { max(Qi((uj )i − (uj )j )) + ∂Fij ∂Fij 2 2 (4.42) +max(Qi((uj )j − (¯uj )j )) + max(uj − u¯j )j ) }≤ ∂Fij ∂Fij

1 2 2 2 2 ≤ C{ k (uj)i − (uj )j kL (Fij ) +max(Qi(uj − u¯j )j ) + max(uj − u¯j )j }. hi ∂Fij Fji 18 Dryja, Galvis and Sarkis

Hence, using a discrete Sobolev inequality, see [34], and the H1/2 stability of the L2 projection we obtain

2 Hi 2 Hi 2 max(Q (u − u¯ ) ) ≤ C(1 + log )|(u ) | 1/2 ≤ C(1 + log ) k ∇(u ) k 2 i j j j j j H (Fji) j j L (Ωj ) ∂Fij hi hi and so

2 1 2 2 (4.43) max((uj )i − (uj )j ) ≤ C{ k (uj )i − (uj )j kL (Fij ) + ∂Fij hi

H 2 + (1+log ) k ∇(uj )j k 2 }. h L (Ωj )

Substituting this into (4.41), we get

2 (4.44) |(u − uˆ ) − (u − uˆ ) | 1/2 ≤ j j i j j j H F (ij )

H 2 2 1 2 2 2 ≤ C(1 + log ) {k ∇(uj )j kL (Ωj ) + k (uj )i − (uj)j kL (Fij )}. h hi

We now estimate the second term of (4.40) as follows. In order to simplify notation we take Fij as the interval [0, H]. Note that

((u − uˆ ) − (u − uˆ ) )2 (4.45) j j i j j j ds ≤ ZFij dist(s,∂Fij ) H/2 ((u − uˆ ) − (u − uˆ ) )2 H ((u − uˆ ) − (u − uˆ ) )2 ≤ C{ j j i j j j ds + j j i j j j ds}. Z0 s ZH/2 (H − s)

Let us estimate the ﬁrst term of right-hand side of (4.45). Let hi ≤ hj (the proof for hi >hj is similar).

H/2 ((u − uˆ ) − (u − uˆ ) )2 hi ((u − uˆ ) − (u − uˆ ) )2 j j i j j j ds = j j i j j j ds + Z0 s Z0 s H/2 2 ((uj − uˆj )i − (uj − uˆj)j ) 2 + ds ≤ C{[(uj − uˆj)i(hi) − (uj − uˆj)j (hi)] + Zhi s Hi 2 +(1 + log )max((uj − uˆj)i − (uj − uˆj)j ) }≤ hi Fij

Hi 2 ≤ C(1 + log )max((uj − uˆj )i − (uj − uˆj )j ) ≤ hi Fij

Hi 2 2 ≤ C(1 + log ){max((uj )i − (uj)j ) + max((ˆuj )i − (ˆuj )j ) }≤ hi Fij Fij

Hi 2 ≤ C(1 + log )max((uj )i − (uj )j ) ≤ hi Fij

H 2 1 2 2 2 2 ≤ C(1 + log ) { k (uj )i − (uj )j kL (Fij ) + k ∇(uj )j kL (Ωj )}. h hi

We have used (4.43) to obtain the last inequality. The second term of the right-hand side of (4.45) is also estimated by the right-hand side of the last inequality. Using FETI-DP for a composite FE/DG discretization 19 this estimate in (4.45) we get ((u − uˆ ) − (u − uˆ ) )2 (4.46) j j i j j j ds ≤ ZFij dist(s,∂Fij )

H 2 2 1 2 2 2 C(1 + log ) {k ∇(uj )j kL (Ωj ) + k (uj )i − (uj )j kL (Fij )}. h hi Substituting (4.44) and (4.46) into (4.40) we get

2 (4.47) k (uj − uˆj)i − (uj − uˆj)j k 1/2 ≤ H00 (Fij )

H 2 2 1 2 2 2 C(1 + log ) {k ∇(uj )j kL (Fij ) + k (uj )i − (uj)j kL (Fij )}. h hi Substituting in turn this and (4.39) for i and j into (4.38) we get

2 H 2 ˜ ˜ (4.48) ρi k (vi)i k 1/2 ≤ C(1 + log ) {di(ui,ui)+ dj (uj,uj )} H00 (Fij ) h

It remains to estimate the second term of the right-hand side of (4.36). The case ∂ 0 Fij where j ∈ Ei is trivial. For the case Fij such that j ∈ Ei we have, see (4.22) - (4.23),

2γ 2 ρiρj ρ k (v ) − (v ) k 2 = × i i i i j L (Fij ) γ γ 2 (ρi + ρj ) 2 2 (4.49) ×k [(ui − uî)i − (uj − uˆj)i] + [(uj − uˆj )j − (ui − uî)j ] kL (Fij )≤ 2 2 2 2 ≤ 2ρi k (ui)i − (ui)j kL (Fij ) +2ρj k (uj )i − (uj )j kL (Fij ) since (ûi)i = (ûj )i and (ûj )j = (ûi)j on Fij and Fji, respectively. Hence

ρiδ 2 2 k (vi)i − (vi)j kL (Fij )≤ 0 lij hij jX∈Ei

δ 2 2 2 2 (4.50) ≤ C {ρi k (ui)i − (ui)j kL (Fij ) +ρj k (uj )i − (uj )j kL (Fij )} 0 lij hij jX∈Ei

≤ C {d˜i(ui,ui)+ d˜j (uj ,uj)}. 0 jX∈Ei Substituting (4.48) and (4.50) into (4.36) we get H (4.51) d˜ (v , v ) ≤ C(1 + log )2{d˜ (u ,u )+ d˜ (u ,u )}. i i i h i i i j j j jX∈Ei Summing the inequalities (4.51) for i from 1 to N and noting that the number of edges of each subdomain can be bounded independently of N, we obtain (4.35) and (4.34). The proof also works with minor modiﬁcations for the case when Fij is a continuous curve of intervals. For that, we should consider discrete Sobolev tools for non straight edges, see for instance [23], and interpret IFij (ui)i and IFji (ui)j as the linear function with respect to parametrized path on the edge deﬁned by the nodal value of (ui)i or (ui)j at x ∈ ∂Fij and ∂Fji. 20 Dryja, Galvis and Sarkis

5. Implementation. The problem (4.15) can be solved efficiently by the preconditioned conjugate gradient method with the preconditioner M defined in (4.27). To simplify the presentation we only discuss the Richardson’s method. For the system of algebraic equations, see (4.15), (5.1) F λ∗ = g the Richardson iterative method is of the form: with given λ0 and for k =0, 1, · · · −1 (5.2) λk+1 = λk − τoptM (F λk − g) 2 where τopt =2/(C(1 + log H/h) + 1)), see (4.28). We need to compute first ˜−1 T ˜−1 (5.3) r˜k := F λk − g = B△S (B△λk − g˜△)= B△S g˜k and then, see (4.27),

N −1 (i) (i) ′ (i) (i) T (5.4) rk := M r˜k = B△ D△ Si,△D△ (B△ ) r˜k Xi=1 N ′ ′ wherer ˜k = {r˜i,k}i=1 ∈ W△(Γ ) withr ˜i,k ∈ Wi,△(Γi).

To computer ˜k we need to solve a system

(5.5) Sv˜ k =˜gk. Note that S˜ is a global matrix but with very weak couplings only through the corners of substructures Ωi for 1 ≤ i ≤ N. Hence, the system with S˜ is solved by a special algorithm based on the Cholesky factorization. For the conforming case this algorithm is described in the book [34], see pp. 166-167. We modify this algorithm to solve (5.5). The main modiﬁcation corresponds to the fact that in our case in a common corner of substructures Ωi we have multiple unknowns while in the conforming case we have only one value. A computation of B△v, see (4.9) and (4.10), for a given N ′ (i) v = {vi}i=1 ∈ W△(Γ ), reduces to multiply the rectangular matrices B△ with entries ′ {0, 1, −1} by the subvector of vi belonging to Wi,△(Γi).

−1 To compute M r˜k, see (5.4), we need to compute for 1 ≤ i ≤ N

′ (i) (i) T ′ (5.6) Si,△D△ (B△ ) r˜k =: Si,△v˜i,k.

(i) (i) T (i) T A computation ofv ˜i,k := D△ (B△ ) r˜k reduces to a multiplication ofr ˜k by (B△ ) (i) and then by the diagonal scaling matrix D△ , see (4.19). In turn, a computation of ′ ′ Si,△v˜i,k is reduced to solving a local problem deﬁned on Γi, see (4.18), with zero val- ′ ′ ues ofv ˜i at the corners Vi in Ωi. These problems involve the solution of the problem ′ ′ ′ Si on each Ωi with Dirichlet data on Γi. We point out that the local problems are independent so they can be solved in parallel.

Finally compute

λk+1 = λk − τoptrk with the computed above rk. FETI-DP for a composite FE/DG discretization 21

6. Numerical experiments. In this section, we present numerical results for solving the linear system (4.15) with the left preconditioner (4.27). We show that the lower and upper bounds of Theorem 4.3 are reﬂected in the numerical tests. In particular we show that the constant C in (4.28) does not depend on hi, hi/hj, Hi, and the jumps of ρi.

Fig. 6.1. A 4 × 4 subdomain partition with 4 × 4 structured local mesh for the black subdomains and 3 × 3 structured local mesh for the red subdomains.

We consider the domain Ω = (0, 1)2 and divide into N = M × M squares subdomains Ωi of size H = 1/M. Inside each subdomain Ωi we generate a structured triangulation with ni subintervals in each coordinate direction and apply the discretization presented in Section 2.2 with penalty term δ = 4. In the numerical experiments we use a red and black checkerboard type of subdomain partition, where the most bottom-left subdomain has a black color. On the black subdomains we let L Lr ni = nb = 2 ∗ 2 b and on the red subdomains we let ni = nr = 3 ∗ 2 , where Lb and Lr are integers denoting the number of refinements inside each subdomain Ωi. See Figure 6.1 for the case M = 4, Lb = 1 and Lr = 0. Hence the local mesh sizes 1 1 are hb = and hr = , respectively. We solve the second order elliptic prob- Mnb Mnr ∗ lem −div(ρ(x)∇uex(x)) = 1 in Ω with homogeneous Dirichlet boundary conditions ∗ u = 0. In the numerical experiments, we run PCG until the l2 initial residual is reduced by a factor of 108. In all the experiments we consider β = 1, see (4.19). In the first test of experiments we consider the constant coefficient case ρ = 1. We consider different values of M × M coarse partitions and different values of local refinements Lb = Lr, therefore keeping constant the mesh ratio hb/hr = 3/2. Table 6.1 lists the number of PCG iterations and in parenthesis the condition number estimate of the preconditioned system. As expected from the analysis, the condition numbers appear to be independent of the number of subdomains and grow by a two- logarithmically factor when the size of the local problems increases. As expected from the theory, the lower bounds estimates are always very closed to one, therefore, we do not show in the tables. We now consider the discontinuous coefficients case where we set ρb = 1 on the black substructures and we vary ρr on the red substructures. The substructures partition is kept fixed to 4 × 4. Table 6.2 lists the results on runs for different values of ρr and for different levels of refinements Lr on the red substructures. On the black substructures nb = 2 is kept fixed. The performance of the preconditioner is robust with respect to the coefficients and the mesh ratio hb/hr as predicted. 22 Dryja, Galvis and Sarkis

Table 6.1 Number of iterations, condition numbers (in parenthesis) for diﬀerent sizes of coarse and local problems and with constant coeﬃcient ρb = ρr = 1 and Lb = Lr.

Lr ↓ M → 2 4 8 16 0 10 (2.22) 12 (2.94) 14 (3.36) 15 (3.50) 1 12 (2.94) 15 (3.88) 16 (4.15) 17 (4.26) 2 14 (3.27) 15 (4.51) 17 (4.86) 18 (4.98) 3 15 (3.34) 15 (5.34) 19 (5.81) 19 (5.94) 4 15 (3.34) 17 (6.39) 21 (6.99) 21 (7.14)

Table 6.2 Number of iterations and condition numbers (in parenthesis) for different values of the coefficient ρr and local meshes with Lr refinements on the red substructures. On black substructures the coefficient ρb = 1 and Lb = 0 are kept fixed. The substructure partition is also kept fixed to 4 × 4.

Lr ↓ ρr → 1000 10 1 0.1 0.001 0 6 (1.18) 9 (1.79) 12 (2.94) 9 (1.91) 5 (1.18) 1 6 (1.15) 10 (1.79) 14 (3.56) 11 (2.30) 5 (1.28) 2 5 (1.13) 10 (2.05) 17 (4.39) 11 (2.47) 5 (1.39) 3 5 (1.11) 11 (2.19) 17 (4.54) 12 (2.44) 5 (1.50) 4 5 (1.11) 11 (2.22) 18 (4.72) 11 (2.33) 5 (1.62)

7. Conclusions. In this paper we consider a composite FE/DG discretization for handling second order elliptic equations with discontinuous coefficients and nonmatching meshes across subdomains. We design and analyze a FETI-DP preconditioner for the associated discrete system and prove (see Theorem 4.3 and Lemmas 4.4 and 4.5) that the condition number of of this preconditioned system only grows two- logarithmically with respect to Hi/hi, and does not depend on local mesh size hi, a size of the subdomains Hi, and jumps of coefficient ρi and the mesh ratio hi/hj across subdomain edges Fij . The numerical tests confirm the theoretical results. Up to how the diagonal weighting matrices (4.19) are chosen and the availability of Neumann matrices of the subdomains, the method is fully algebraic. The algorithm can also be extended to 3-D cases by selecting primals which impose continuity at the corners of subdomains and on averages across the edges of subdomains, and its mathematical analysis needs a lot of details, therefore, it is not discussed in the paper. It will be considered separately. The algorithm also can be extended to full DG discretization, and it will be discussed elsewhere. Acknowledgment. We would like to express our thanks to the two anonymous referees for the suggestions to improve the presentation of the paper.

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