November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

Mathematical Models and Methods in Applied Sciences c World Scientiﬁc Publishing Company

Isogeometric BDDC Deluxe preconditioners for linear elasticity

L. F. Pavarino˚ Dipartimento di Matematica, Universit`adegli Studi di Pavia, Via Ferrata 5, 27100 Pavia, Italy. [email protected]

S. Scacchi˚ Dipartimento di Matematica, Universit`adegli Studi di Milano, Via Saldini 50, 20133 Milano, Italy. [email protected]

O. B. Widlund: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA. [email protected]

S. Zampini Extreme Computing Research Center, King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia. [email protected]

TR2017-988

Balancing domain decomposition by constraints (BDDC) preconditioners have been shown to provide rapidly convergent preconditioned conjugate gradient methods for solving many of the very ill-conditioned systems of algebraic equations which often arise in ﬁnite element approximations of a large variety of problems in continuum mechanics. These algorithms have also been developed successfully for problems arising in isogeometric analysis. In particular, the BDDC deluxe version has proven very successful for problems approximated by non-uniform rational B-splines (NURBS), even those of high order and regularity. One main purpose of this paper is to extend the theory, previously fully developed only for scalar elliptic problems in the plane, to problems of linear elasticity in three dimensions. Numerical experiments supporting the theory, are also reported. Some of these experiments highlight the fact that the development of the theory can help to decrease substantially the dimension of the primal space of the BDDC algorithm, which provides the necessary global component of these preconditioners, while maintaining scalability and good convergence rates.

Keywords: domain decomposition; BDDC deluxe preconditioners; isogeometric analysis;

˚This work was supported by grants of M.I.U.R. (PRIN 201289A4LX 002) and of Istituto Nazionale di Alta Matematica (INDAM-GNCS). :This work has been supported by the National Science Foundation Grant DMS-1522736

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2 L. F. Pavarino, S. Scacchi, O. B. Widlund, S. Zampini

NURBS; compressible linear elasticity.

AMS Subject Classiﬁcation: 65F08, 65N30, 65N35, 65N55

1. Introduction Isogeometric Analysis (IGA) is a recent technology, introduced in 34, for the numerical approximation of Partial Differential Equations (PDEs), where the description of the domain of the PDE is adopted from a Computer Aided Design (CAD) parametrization usually based on Non-Uniform Rational B-Splines (NURBS); see the monograph 20. In IGA, NURBS basis functions are used not only to represent the CAD geometry, but also as basis functions for Galerkin approximations, using an isoparametric paradigm. Since their introduction, IGA techniques have been studied and applied in several different fields, e.g., structural mechanics 2, fluid dynamics 4, electromagnetics 15 and computational electrocardiology 17,46. Conver- gence analysis was first developed in 5; see also the recent review 6. NURBS basis functions have several interesting properties such as local support, positivity, providing a partition of unity, and a variation diminishing property. The ease of IGA in building spaces with high inter-element regularity allows for quite small problems (in terms of degrees of freedom) compared with standard finite element methods. However, IGA discrete problems might still be very large in realistic problems of interest, and their condition numbers grow quickly with the inverse mesh size h´1 and/or the polynomial degree p. In addition, the presence of discon- tinuities in the material parameters can further increase the condition numbers. As a consequence, in both the mathematical and the engineering communities, there is a growing interest in developing efficient preconditioners for IGA discrete problems. Since 2012, several iterative solvers have been developed for IGA discretizations of different families of PDEs: see 19 for ILU and Block Jacobi preconditioners, 7,10,9 for Overlapping Additive Schwarz, 41,32,47 for Finite Elements Tearing and Interconnecting Dual Primal methods (FETI-DP), 14 for BPX, 49 for Sylvester equation-based preconditioners and 31,28,27,33 for Multigrid methods. A main purpose of this article is to extend the theory previously developed for Balancing Domain Decomposition by Constraints (BDDC) preconditioners for scalar elliptic problems in the plane to isogeometric analysis for three-dimensional elasticity. This earlier theory was first developed in 8 and later applied in a first study of the BDDC deluxe for isogeometric analysis, see 11, and in a study of the adaptive selection of the primal space of BDDC deluxe spaces, see 12. Fundamentally, many domain decomposition proofs are exercises in finding a stable decomposition of an arbitrary element in the discrete space into elements of subspaces, often directly associated with geometric objects. In the present context, a major role is played by the equivalence classes of the decomposition of the original global domain into non-overlapping subdomains. In isogeometric analysis these equivalence classes are associated with a subset of knots of a fat interface. The fat interface is defined by those knots of the B-spline basis functions with the interior November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

Isogeometric BDDC Deluxe preconditioners for linear elasticity 3

of their supports intersecting the thin interface obtained by a traditional decomposition of the domain into subdomains. We can then decompose this interface into fat subdomain faces, edges, and vertices. We will be carrying out our analysis in the parametric domain where these subdomains are cuboids, i.e. parallelepipeds with rectangular faces. Equivalence classes are given in terms of the indices of the subdomains to which the knots belong. It is well known that successful domain decomposition algorithms all require a coarse, global component. For BDDC and FETI–DP algorithms, the global component is directly associated with primal variables defined by a subspace of interface variables. Given that the fat interfaces typically have a larger dimension than com- parable finite element interfaces, it is quite important to be very selective in choosing the primal variables. In this, we will be able to learn a great deal from earlier work on the finite element case, in particular from 40. The rest of the paper is organized as follows: in the next section, we introduce the equations of linear compressible elasticity in three dimensions and their approximation using NURBS. Following 8, we also show that it is possible to carry out the analysis using a Q1 finite element space. In Section 3, the fat interface set and the equivalence classes that partition this set are introduced followed by a description of the BDDC algorithms, in particular, its deluxe variant. Sets of primal constraints are introduced in Section 4; the search for effective small sets of primal constraints is inspired by earlier work on lower order finite elements as reported in 37,38,40. We prove that in many cases only primal constraints on the averages of the three components of the displacement over just one thin edge for each fat interface edge are required while for some distributions of the Laméparameters of the subdomains, additional constraints, including first order moments of the displacement as well as primal constraints associated with the fat vertices, will be required. In this, we distinguish between the case of quasi-monotone coefficients and a general case; see 29 for an introduction of quasi-monotonicity. In Section 5, we report on numerical experiments which support and supplement the theory and in a final section, we offer some conclusions.

2. Isogeometric discretization of the linear elasticity system We will consider the linear elastic deformation of a material body Ω, that is a domain in R3, represented exactly by the isogeometric analysis system. Its boundary BΩ is divided into two nonoverlapping subsets ΓD and ΓN . The body is clamped along 3 ΓD and subject to a given traction g2 :ΓN Ñ R on ΓN , as well as to a body force 3 3 of density g1 :Ω Ñ R . The displacement ﬁeld u :Ω Ñ R , describing the linear elastic deformation of Ω, is the solution of the system: November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

4 L. F. Pavarino, S. Scacchi, O. B. Widlund, S. Zampini

div Cεpuq ` g1 “ 0 in Ω

$ u “ 0 on ΓD (2.1) ’ &’ Cεpuq ¨ n “ g2 on ΓN , ’ where ε denotes the symmetric%’ gradient operator and n the unit outward normal of the boundary. The fourth order tensor C is defined by Cτ :“ 2µτ ` λtrpτ qI (2.2) for any second order tensors τ , where trpτ q is the trace of τ , and λ and µ are 2µν the Laméparameters. λ “ 1´2ν is defined in terms of the strictly positive shear modulus µ and the Poisson’s ratio ν, satisfying 0 ď ν “ νpxq ă 1{2 (compressible materials). 2 3 2 3 Assuming, for simplicity, regular loads, i.e., g1 P rL pΩqs and g2 P rL pΓN qs , we introduce 1 1 3 ă f, v ą:“ pg1, vqΩ ` pg2, vqΓN @v P H pΩq :“ rH pΩqs , (2.3) 2 where p¨, ¨qΩ and p¨, ¨qΓN are the L scalar products over Ω and ΓN , respectively. The variational formulation of problem (2.1) then reads:

1 Find u P HΓ pΩq such that: D (2.4) a u, v f, v v H1 Ω , # p q “ă ą @ P ΓD p q where H1 Ω v H1 Ω v 0 and ΓD p q “ t P p q | |ΓD “ u

a w, v : 2µ ε w : ε v dx λ div w div v dx w, v H1 Ω , (2.5) p q “ p q p q ` @ P ΓD p q żΩ żΩ with the symbol : denoting the standard contraction operator. When developing the theory, we will assume that the Lam´eparameters are constant in each subdomain but will allow jumps in their values across the interface between the subdomains.

2.1. Isogeometric discretization p Given univariate B-spline basis functions Li pξq of degree p associated to the knot vector tξ1 “ 0, ..., ξ``p`1 “ 1u defined on the parametric interval I :“ p0, 1q, and q r similarly defined univariate functions Mj pηq and Nk pζq, we define by a tensor product the 3D parametric space on Ω:“ p0, 1q ˆ p0, 1q ˆ p0, 1q, the `pˆ m ˆ n mesh of control points Ci,j,k associated with the knot vectors tξ1 “ 0, ..., ξ``p`1 “ 1u, tη1 “ 0, ..., ηm`q`1 “ 1u, tζ1 “p0, ..., ζn`r`1 “ 1u, the trivariate B-spline basis p,q,r p q r functions by Bi,j,k pξ, η, ζq “ Li pξq Mj pηq Nk pζq, and the trivariate B-spline discrete space as p,q,r Sh :“ spantBi,j,k pξ, η, ζq, i “ 1, . . . , `, j “ 1, . . . , m, k “ 1, . . . , nu. (2.6)

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Isogeometric BDDC Deluxe preconditioners for linear elasticity 5

Analogously, the NURBS space is the span of NURBS basis functions defined in one dimension by p p p Li pξqωi Li pξqωi Ri pξq :“ “ , (2.7) ` Lp ξ ω wpξq î“1 î p q î with the weight function w ξ : ř` Lp ξ ω S , and in three dimensions by a p q “ î“1 î p q î P h tensor product ř p,q,r p p,q,r p,q,r Bi,j,k pξ, η, ζqωi,j,k Bi,j,k pξ, η, ζqωi,j,k Ri,j,k pξ, η, ζq :“ “ , (2.8) ` m n Bp,q,rpξ, η, ζqω wpξ, η, ζq î“1 ˆj“1 kˆ“1 î,ˆj,kˆ î,ˆj,kˆ

where wpξ, η, ζq isř the weightř ř function and ωî,ˆj,kˆ are positive weights associated with a ` ˆ m ˆ n net of control points. The discrete NURBS space on Ω is then defined as the span of the push-forward of the NURBS basis functions (2.8), i.e. p,q,r ´1 Nh :“ spantRi,j,k ˝ F , with i “ 1, . . . , `, j “ 1, . . . , m, k “ 1, . . . , nu, (2.9) with F : Ω Ñ Ω, the geometrical map between the parameter and physical spaces ` m n p,q,r given by Fpξ, ηq “ i“1 j“1 k“1 Ri,j,k pξ, η, ζqCi,j,k. The spline space in the p 1 parameter space is then defined as Vh :“ Sh X H pΩq and the NURBS space in ř ř ř ΓD the physical space as U : N H1 Ω . The IGA formulation of problem (2.4) h “ h X ΓD p q then reads: find uh P Uh such that: p p p

apuh, vhq “ă f, vh ą @v P Uh. (2.10) The matrix form of (2.10) is the linear system

Auh “ fh, (2.11) with a symmetric positive deﬁnite stiﬀness matrix A.

2.2. Preliminaries and Q1 ﬁnite elements In this subsection, we will ﬁrst follow 8 pSection 5q quite closely; no new ideas are required when we step from the two-dimensional scalar case to the vector valued in three dimensions. We will be working with tensor products of B-splines but note that the same results carry over to the non-uniform rational B-splines (NURBS), that span the isogeometric analysis spaces, provided that the geometric maps are well behaved. Let Ωlmn :“ Il ˆ Im ˆ In, be a generic subdomain in the parametric space built

by a tensor product of three intervals. Here Il “ pξil , ξil`1 q and ξil , ξil`1 are knots associatedp with thep spacep ofp B-splines. We note that there exist l1 ď l2 ă l3 ď l4 such p that the univariate B-spline basis functions ptLi pξq, i “ l1, . . . , l4u, with supports intersecting Il, can be separated into

p l2 p tL pξqu such that ξi P supppL pξqq, (2.12) p i i“l1 l i Lp ξ l3´1 such that supp Lp ξ I , (2.13) t i p qui“l2`1 p i p qq Ď l p l4 p tL pξqu such that ξi P supppL pξqq, (2.14) i i“l3 l`1 i p November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

6 L. F. Pavarino, S. Scacchi, O. B. Widlund, S. Zampini

FATΘFAT(k) FACE FACEF

Ω(k) i

FATΘ(k) VERTEXC

FAT(k) EDGEΘ E

Fig. 1. Schematic illustration of a subdomain Ωi in the parametric space, with spline polynomial degree p “ 3 and regularity k “ 2, displaying a fat vertex (with pk ` 1q3 knots), a fat edge (with pk ` 1q2 thin edges), and a fat face (with k ` 1 thin faces).

with boundary subdomains being treated similarly. The fact that l1 often diﬀers from l2 and l3 from l4, etc., will result in fat interfaces. By analogous decompositions of the basis functions in the other two dimensions, p there exist indices m1 ď m2 ă m3 ď m4 for tMj pηq, j “ m1, . . . , m4u associated p with Im “ pηjm , ηjm`1 q, and n1 ď n2 ă n3 ď n4 for tNk pρq, k “ n1, . . . , n4u asso-

ciated with In “ pρjn , ρjn`1 q, with the same properties as the first set. A function p z P V restricted to Ωlmn can therefore be represented by p l4 m4 n4 p p p p zpξ, η, ρpq “ cijkL pξqM pηqN pρq . (2.15) Ωlmn i j k Ωlmn i“l1 j“m1 k“n1 ˇ ÿ ÿ ÿ ˇ ˇ p ˇ p Here the cijk are 3´vectors; we will denote their `2ńorms by ˇ|cijk|. We will always assume that the knots are quasi-uniformly spaced with the distance between different, consecutive knots in the coordinate direction denoted by h. We will also assume that the elements and the subdomains are shape-regular. We will also use the same notation for vectors that represent functions such as z. Centered around the faces, edges, and vertices of the non-overlapping subdomains into which the original parametric space has been subdivided, we will have fat faces, fat edges, and fat vertices. These sets do not intersect. A fat face can be viewed as being built from a set of parallel thin faces and a fat edge from a set of parallel thin edges all parallel to the long side of the fat edge. In what follows, we will indicate the set of indices, introduced above, by

θ :“ tpi, j, kq : l1 ď i ď l4, m1 ď j ď m4, n1 ď k ď n4u, (2.16)

and denoting by K an element contained in Ωlmn, we deﬁne θpKq :“ tpi, j, kq : K Ď supppLppξqM ppηqqN ppρqqu. (2.17) p i j k November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

Isogeometric BDDC Deluxe preconditioners for linear elasticity 7

For any z P V , we deﬁne the local and global discrete norms Ωlmn

2 ˇ p 2 3 2 2 3 |z|K “pˇ max |cijk| h , |z|l :“ |cijk| h , pi,j,kqPθpKq i,j,k θ K ´ ¯ p ÿqP p q 2 2 2 2 2 2 |z|∇ :“ |z|ξ ` |z|η ` |z|ρ, where |z|ξ :“ |cijk ´ ci´1,jk| h, i,j,k θ ξ p ÿqP p q

and θpξq :“ tpi, j, kq : l1 ` 1 ď i ď l4, m1 ď j ď m4, n1 ď k ď n4u and analogously 2 2 for |z|η and |z|ρ. By using the tensor product structure of the B-spline basis functions, we can directly extend the 2D proofs of 8 pP rop. 5.1, 5.2, Corol. 5.1q to the 3D case to obtain the following results.

Proposition 2.1. Let K be an element contained in Ωlmn. Then for any z P V Ωlmn

}z}L2pKq « |z|K , |z|l «}z} 2 , |z|∇ « |z| 1 . ˇ p L pΩlmnq p H pΩlmnq pˇ

Here }z}L2pKq « |z|K means that therep are positive constantspc and C, such that

c}z}L2pKq ď |z|K ď C}z}L2pKq, etc. We also deﬁne the L ˆ M ˆ N real-valued tensors C with the semi-norm

L M N L Mr N 2 2 r 2 C ∇ :“ |cijk ´ ci´1,jk| ` |cijk ´ ci,j´1,k| i“2 j“1 k“1 i“1 j“2 k“1 ˇˇˇ ˇˇˇ ´ ÿ ÿ ÿ ÿ ÿ ÿ (2.18) ˇˇˇ r ˇˇˇ L M N r 2 ` |cijk ´ ci,j,k´1| h, i“1 j“1 k“2 ÿ ÿ ÿ ¯ where L “ l4 ´l1 `1, M “ m4 ´m1 `1 and N “ n4 ń1 `1. This effectively provides a H1´semi-norm for the vector valued function z defined in formula p2.15q; cf. 8 pSubsec. 5.1.1q. We note that the tensor C can be viewed as a mesh function defined on a cuboid and that, when developing the theory, we can work exclusively with r the Q1 finite element space.

3. Equivalence classes, Schur complements, and BDDC preconditioners Balancing Domain Decomposition by Constraints (BDDC) preconditioners have evolved from the balancing Neumann-Neumann methods. They were introduced by Dohrmann in 22 and first analyzed by Mandel, Dohrmann, and Tezaur in 43,44. All the local and the coarse problems are non-singular because of a choice of sufficiently many primal continuity constraints across the interface of the subdomains. Another advantage is that the coarse and all the local problems can be solved in parallel. The primal constraints can be point constraints and/or, more effectively, averages November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

8 L. F. Pavarino, S. Scacchi, O. B. Widlund, S. Zampini

or moments over edges or faces of the subdomains. We refer to 55 for a recent review about the BDDC method and for additional references. We refer to the Toselli and Widlund 50 pCh. 6q for a detailed discussion of Neumann-Neumann, FETI and FETI-DP algorithms, see also 30,40,42. A FETI- DP and a BDDC algorithm, with the same set of primal constraints, are closely related and the two relevant operators have the same spectra, except for possible eigenvalues at 0 and 1, see 44. BDDC algorithms belong to the class of non-overlapping domain decomposition algorithms, being based on decompositions of the domain Ω of an elliptic operator into non-overlapping subdomains Ωi. The subdomain interfaces Γi of Ωi do not cut through any elements and are defined by Γi :“BΩizBΩ. In the isogeometric context, the subdomains Ωi are images of cuboids Ωi in the parameter space, each a union of hexahedral elements defined by the eight knots which are its vertices. Associated with each subdomain interfacep is part of a fat interface and a set of equivalence classes. For three dimensional problems, the equivalence classes associated with a subdomain are defined as follows: we first separate the knots of

the interior of the subdomains and those associated with the interface Γ :“ i Γi; those in the interior are the knots with B-spline basis functions supported in the Ť closure of individual subdomains. The set of the remaining, the interface knots, are partitioned into equivalence classes associated with subdomain vertices, edges, and faces. Thus, we separate off the vertex equivalence sets, which are given by the knots with B-spline basis functions with a subdomain vertex in the interior of their supports. We next identify the edge equivalence classes among the remaining interface knots with B-spline basis functions with supports that intersect a subdomain edge in its interior. Finally, the remaining interface knots, which have basis functions with supports intersecting a subdomain face, are separated into subsets associated with the individual subdomain faces. What has just been said is valid for subdomains in the interior. If a Dirichlet condition is imposed on a face of a subdomain, its degrees of freedom are eliminated and do not enter the system. If Neumann conditions are given for a subdomain face, its knots are assigned to the interior set of the subdomain. If two or more adjacent subdomains touch the Neumann boundary of the domain Omega, then the knots on the boundary faces are again interior while its edge and vertex knots touching the Neumann boundary are now shared between several subdomains and belong to the fat interface. A boundary fat edge shared by two subdomains, instead of four as in the interior of Omega, should then be considered part of an extended face shared by those two subdomains, a face orthogonal to the Neumann boundary; these knots naturally belong to the same equivalence class as those on that fat face. This is fully consistent with the definition of the equivalence classes. An analogous rule is used for a fat boundary vertex. Once these equivalence classes have been identified, we will find many similarities with the development of BDDC algorithms for finite element problems. November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

Isogeometric BDDC Deluxe preconditioners for linear elasticity 9

piq Given the stiﬀness matrix A of the subdomain Ωi, we obtain a subdomain Schur complement Spiq by eliminating the interior variables, i.e., all those associated with the basis functions with supports conﬁned to Ωi. We will also work with principal minors of these Schur complements associated with a fat subdomain piq piq piq vertex, edge, and face, denoting them by SVV ,SEE, and SFF , respectively. The interface space is then divided into a primal subspace WΠ of functions which are continuous and a complementary dual subspace W∆ for which we will allow multiple values across the interface during part of the iteration. The BDDC and FETI–DP algorithms can be described in terms of three product spaces of functions associated with subsets of interface knots:

WΓ Ă WΓ Ă WΓ. W is built as a product space of components associated with the individual Ω , Γ x | i without any continuity constraints across the interface. Elements of WΓ have common values of the primal variables but allow multiple values of the dual variables while the elements of WΓ are continuous at all knots. After eliminating| the interior variables and, if need be, changing the variables, we can then write the subdomain Schur complements asx

piq piq piq S∆∆ S∆Π S “ piq piq . ˜ SΠ∆ SΠΠ ¸ We will partially subassemble the Spiq, obtaining S, enforcing the continuity of the primal variables only. Thus, we then work in WΓ. In each step of the iteration, we will solve a linear system with the coefficient matrixq S. Alternatively, we could also work with a linear system with a matrix obtained| by partially subassembling the subdomain stiffness matrices Apiq. We note that solvingq these linear systems will be considerably much faster than working with the fully assembled system if the dimension of the primal space is modest. At the end of each iteration, the approximate solution is made continuous at all knots of the interface, by applying a weighted averaging operator ED, which maps WΓ into WΓ. In each step of the iteration, we first compute the residual of the fully assembled T Schur complement. We then apply ED to obtain| a right-handx side of the partially subassembled linear system, solve this system and then apply ED. This last step changes the values on Γ, unless the iteration has converged, and can result in non- zero residuals at interior knots next to Γ. In a final step of each iteration step, we eliminate these residuals by solving a Dirichlet problem on each of the subdomains. However, for the deluxe variant, this last step will not be needed. We always accel- erate the iteration with the preconditioned conjugate gradient (PCG) algorithm.

3.1. The deluxe scaling What is crucial in designing eﬀective BDDC algorithms is the good choice of a set of primal constraints and a good recipe for averaging across the interface. In this November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

10 L. F. Pavarino, S. Scacchi, O. B. Widlund, S. Zampini

paper, we will always use the deluxe recipe in the construction of the averaging operator ED; in our previous work on BDDC for IGA, we have found this choice much superior to the alternatives that we also have tried; see 11,12. The deluxe variant originates in studies on three-dimensional Hpcurlq and Hpdiv q problems in which it was found that traditional averaging recipes did not work uniformly well, cf. 45,25,26. The reason of such occasional failure is the fact that there are two sets of material parameters in these applications. The deluxe variant, that was then introduced, has proven quite successful in a variety of applications; see, e.g., 52,11,54,55,56,57. We also note that this variant plays an important role in the development of algorithms for which the primal space is chosen adaptively; see, in particular, 24. A face component of the average operator ED, for a problem in three dimensions, ij across a fat subdomain face F , common to two subdomains Ωi and Ωj, is deﬁned in terms of the equivalence set of variables of that face and the corresponding principal pkq pkq ij minors SFF of the S , k “ i, j. The deluxe averaging operator, for F , is then deﬁned by

piq pjq ´1 piq piq pjq pjq w¯F ij :“ pEDwqF ij :“ pSFF ` SFF q pSFF wF ` SFF wF q.

piq piq ij pjq Here wF is the restriction of w to the face set, F , (and analogously for wF ). By exchanging F by E, we obtain the formula for an edge for a 2D problem. piq pjq ´1 The action of pSFF ` SFF q can be implemented by solving a Dirichlet prob- ij lem, with zero boundary values, on Ωi Y F Y Ωj, with a right hand side which vanishes in the interiors of the two subdomains. This can add signiﬁcantly to the cost. However, we note that some software systems for the parallel solution of systems of linear algebraic equations, such as MUMPS, see 1, provide the subdomain Schur complement matrices Spiq. In the economic version (e-version), we replace this large domain by a thin domain built from one or a few layers of elements next to the face and this often results in a very similar performance; see, e.g., 26,56. Deluxe averaging operators are also developed for subdomain edges and subdomain vertices for problems in three dimensions. Given the simple geometry of the parameter space that we are considering, we ﬁnd that in all these cases the equivalence classes will have four and eight elements for any interior fat subdomain edge and vertex, respectively. Thus, for an interior fat subdomain edge E in 3D, shared by subdomains Ωi, Ωj, Ωk, and Ω`, we can use the formula

piq pjq pkq p`q ´1 piq piq pjq pjq pkq pkq p`q p`q w¯E :“ pSEE ` SEE ` SEE ` SEEq pSEEwE ` SEEwE ` SEEwE ` SEEwE q.

An analogous formula holds for the interior fat vertices and involves eight operators. The core of any estimate for a BDDC algorithm involves the S´norm of the average operator ED. By an algebraic argument known, for FETI–DP, since 2002, cf. 39, we have the following bound for the condition number: q

´1 κpM Aq ď }ED}S. (3.1)

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Isogeometric BDDC Deluxe preconditioners for linear elasticity 11

Here M ´1 represents the action of the preconditioner. We recall that the relevant operators of FETI-DP and BDDC methods with the same set of primal constraints have the same spectrum, except for possible eigenvalues at 0 and 1, see 44,42,13. For FETI–DP, the relevant operator is PD :“ I ´ ED. In the development of theory for both FETI-DP and BDDC, it is convenient to work with PD rather than ED and we will do so in this paper as well. In fact, the two operators have the same norm, see 24 pAppendix Eq.

4. Primal constraints for three-dimensional elasticity and main results In this section, we will develop our theory which is based, in part, on earlier work in 8 and 40. In the first of these papers, it was shown that much of the analysis can be reduced to a study of related finite element problems, cf. subsection 2.2. In the second paper, a framework was introduced which allows for the development of small primal spaces for lower order finite element approximations of compressible elasticity. For isogeometric problems, we will often have fat interfaces and therefore potentially even much larger primal spaces than for finite element problems and our efforts will be focused on keeping the dimension of these global spaces small.

4.1. Primal constraints As demonstrated in 11 and 16, the deluxe version of BDDC often has the advantage that the analysis of bounds necessary for a condition number bound can be reduced to bounds over individual subdomains; this is true of the analysis in subsection 4.3 but not of the one in subsection 4.2. When developing primal constraints, we will, as in previous work on finite elements, find that averages and first order moments over thin edges are particularly useful; see 40. Thus, we will work with individual thin edges, that are parts of the fat edges, in fact with just one such edge for each fat subdomain edge and with only the averages over the three displacement components when constructing primal constraints. In some cases, we will also work with fully primal edges defined, for a thin edge E, as in 40 pDef. 5.4q, in terms of primal constraints given by

w ¨ rids gE pwq :“ E , i “ 1,..., 6, (4.1) i r ¨ r ds şE i i where the ri are the standard basisş functions of RB, the space of rigid body modes. On a thin edge E which is part of the x3áxis and with the origin at its midpoint, T T T T we choose r1 “ p1, 0, 0q , r2 “ p0, 1, 0q , r3 “ p0, 0, 1q , r4 “ p1{Hqp0, x3, ´x2q , T T r5 “ p1{Hqpx3, 0, ´x1q , r6 “ p1{Hqp´x2, x1, 0q ; note that r6 “ 0 on the edge E and that g6 then serves no useful purpose. The factors p1{Hq are introduced to make the L2pEqńorm of the first five of these vectors be of the same order of magnitude. The primal constraints for a fully primal edge are then given by E pjq E pkq gi pw q “ gi pw q, 1 ď i ď 5, for all relevant pairs of j and k. In this coordinate November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

12 L. F. Pavarino, S. Scacchi, O. B. Widlund, S. Zampini

system, we can use these functionals to derive a representation of any rigid body mode and any x P E, 5 E r “ gi prqri, @r P RB. (4.2) i“1 ÿ 2 This follows from the fact that the ri are orthogonal in L p´H{2,H{2q and that E gi prjq “ δij for 1 ď i, j ď 5. We also use that r6 “ 0 on the edge E. By moving the origin of our coordinate system and rotating it, we can always E make g6 pwq vanish identically for any thin edge; we will always work in such coordinate systems when developing our theory. For many problems, we will only need to use the three functionals associated E E with the translations in the three coordinate directions, i.e., only work with g1 , g2 , E and g3 . We note that our choice of primal constraints are then the same as those of 37, an experimental study of the performance of the FETI–DP algorithm for lower order finite element approximations where the Laméparameters are constant or vary slowly in the entire domain Ω. In such cases no additional primal constraints directly associated with subdomain faces and vertices are needed. In our study, we will also confine ourselves to choosing primal constraints for just one thin edge for each fat edge. As demonstrated in 40 pP rop. 5.1q, we can develop an alternative representation to (4.2) of all rigid body modes by working with three such linear functionals for each of the four fat edges that are adjacent to a fat face F . Six functionals are then generated as linear combinations of these twelve functionals providing a dual basis F 6 F tfi ui“1, i.e., fi prjq “ δij, and the representation 6 F r “ fi prqri, @r P RB. (4.3) i“1 ÿ We note that it has been established that the coefficients expressing the six linear functionals in terms of the twelve are all of order 1; see 40 ppp. 1542´3q. In contrast to (4.2), this formula holds for all x. We are also free to select any orthogonal basis triu for RB, in particular the one introduced above for a thin edge E. As a consequence of defining the primal constraints in terms of the linear func- E E E tionals g1 , g2 , and g3 when constructing primal constraints for one thin edge for F piq each of the four fat edges that borders the face, we can conclude that fk pw q “ F pjq fk pw q, k “ 1,..., 6. Useful bounds for these linear functionals are collected in 50 pDef. 5.3q: gE w 2 CH´1 1 log H h w 2 , (4.4) | i p q| ď p ` p { qq} }H1pΩˆq f F w 2 CH´1 1 log H h w 2 . (4.5) | i p q| ď p ` p { qq} }H1pΩˆq

Here H{h, as usual in the domain decomposition literature, is short for maxi Hi{hi, where Hi is the subdomain diameter and hi the characteristic mesh size for the i-th subdomain. These bounds can easily be established by using 50pLemma 4.16q. We will November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

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use these functionals and bounds when estimating the contributions to our bound of the norm of PDw from the fat faces, edges, and vertices. Following 40 pDef. 5.3q, a subdomain face F will be fully primal if there is a dual F 6 basis tfk u1, which satisfies (4.3) and (4.5). When developing our theory, we will make sure that all subdomain faces are fully primal. As already shown, we can effectively work with Q1 finite elements on a quasi- uniform hexagonal mesh that partitions a cuboid. We can do most of the theoretical work with these finite element functions and using the H1pΩˆq´semi-norm and norm because the bilinear form given by (2.5) is continuous in H1pΩˆq with a bound proportional to the Laméparameter µ. And we can return from the H1pΩˆq´ inner product to the bilinear form by using Korn inequalities, in particular 18pT h. 6.15´1q: there exists a constant C “ CpΩˆq, invariant under dilation, such that,

2 2 2 2 2 |w| 1 ˆ ` p1{H q}w| 2 ˆ ď C εpwq : εpwq dx ` p1{H q}w} 2 ˆ (4.6) H pΩq L pΩq ˆ L pΩq żΩ ` ˘ ˘ and 18 pT h. 6.15´3q: there exists a constant C “ CpΩˆq, invariant under dilation, such that,

2 2 inf }w ´ r} 2 ˆ ď CH εpwq : εpwq dx. (4.7) rPRB L pΩq ˆ żΩ The latter result essentially replaces Poincar´e’sinequality. Here H is the diameter of Ωˆ. For subdomains with boundaries that intersect ΓD, the Korn inequalities need to be replaced by Friedrichs-type inequalities. There is no diﬃculties if a full face is subject to a Dirichlet condition. The situation is more complicated if such a condition is imposed only on one subdomain edge since one rotational rigid body mode will not be controlled by the Dirichlet condition alone which then needs to be complemented by a suitable primal constraint.

4.2. Quasi-monotone coeﬃcients and acceptable paths through faces We begin this section by introducing acceptable edge paths and acceptable vertex paths; cf. 40pDef. 5.6, 5.7q. (We note that acceptable edge paths are called acceptable face paths in that paper.)

Deﬁnition 4.1. (Acceptable edge path) Consider a pair of subdomains Ωi and Ωk which have at least a subdomain edge in common. Then there is an acceptable edge path between them if either they have a face in common or there is a third subdomain Ωj sharing that edge with Ωi and Ωk, with TOL‹µj ě minpµi, µkq, and with a path from Ωi to Ωk which passes through subdomain faces and through Ωj. Here TOL is assumed to be a tolerance of relatively modest size.

Deﬁnition 4.2. (Acceptable vertex path) Consider a pair of subdomains Ωi and Ωk which have at least a subdomain vertex in common. Then there is an acceptable November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

14 L. F. Pavarino, S. Scacchi, O. B. Widlund, S. Zampini

vertex path between them if either they have a face in common or there are one or two other subdomains sharing the vertex with Ωi and Ωk, with TOL ‹ µj ě hi minpµi, µkq, and with a path from Ωi to Ωk through subdomain faces and through Hi that or those two subdomains, which satisfy this condition. Here TOL is assumed to be a tolerance of relatively modest size.

If each fat edge is associated with an acceptable edge path for any relevant pair of subdomains, and each fat vertex is associated with an acceptable vertex path for any relevant pair of subdomains, then the set of Lam´eparameters tµiu are said to be quasi-monotone. We are now ready to formulate and prove the following main result.

Theorem 4.1. Assume that the Lam´eparameters µi are quasi-monotone and that the primal space is spanned by three average displacement constraints for one thin edge of each fat edge. Then the condition number of the BDDC deluxe algorithm satisﬁes

H 3 κpM ´1Aq ď }E } ď CTOL 1 ` log , D S h ˆ ˆ ˙˙ q with C independent of the number of subdomains, the subdomains diameters Hi, the mesh sizes hi, and the elastic coeﬃcients. C could grow with the number of knots across the fat interface.

Proof. We will separately bound the face, edge, and vertex contributions to the norm of PDw. Face contributions. We need only strictly local arguments for individual subdomains and will require essentially no assumptions on the Laméparameters in addition to being quasi-monotone. F ij 6 We will use the six linear functionals for each fat subdomain face, tfk u1, which are the linear combinations of the averages of the components of the displacements chosen above, and which form a dual basis of RB and which satisfies (4.3) and (4.5). ij When considering the restriction of PDw to F , the fat face common to the subdomains Ωi and Ωj, we find that for the deluxe variant of BDDC, the square of the energy-norm of PDpwq|F ij equals

piq pjq T piq pjq piq pjq pwF ´ wF q SFF : SFF pwF ´ wF q; (4.8)

16 piq pjq see, e.g., . Here SFF : SFF is the parallel sum

piq pjq piq´1 pjq´1 ´1 SFF : SFF :“ pSFF ` SFF q of principal minors of Spiq and Spjq, respectively, for the variables associated with the common fat face F ij. Spiq and Spjq are the Schur complements obtained by eliminating all variables of the two subdomains, having F ij in common, except those on the fat interfaces, from the subdomain matrices. November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

Isogeometric BDDC Deluxe preconditioners for linear elasticity 15

Given our primal constraints, we ﬁnd that

6 6 F ij piq F ij pjq fk pw qrk “ fk pw qrk; (4.9) k“1 k“1 ÿ ÿ similar identities will be used in other parts of our proofs. 40 piq pjq By using (4.3), (4.9), and ideas from , we can now express wF ´ wF as 6 6 piq piq F ij piq piq pjq pjq F ij pjq pjq wF ´r ´ fk pwF ´r qrk ´ wF ´r ´ fk pwF ´r qrk (4.10) k“1 k“1 ` ÿ ˘ ` ÿ ˘ piq pjq piq pjq for any r and r P RB. We now use that the parallel sum SFF : SFF is bounded piq pjq from above by SFF as well SFF . We will therefore develop bounds of the norms of pkq pkq wF ´ r , for k “ i and j, i.e., of

pkq pkq T pkq pkq pkq pwF ´ r q SFF pwF ´ r q. (4.11) The remaining terms of (4.10) can be bounded by using (4.5) and 40pLemma 7.1q, which is a simple extension of a standard face lemma 50 pLemma 4.25q. Each of these two bounds contribute a factor p1 ` logpH{hqq and no new ideas are needed. Returning to the first set of terms, i.e., the estimate of the expressions in (4.11), we first note that for the finite element case a good estimate is given by a standard face lemma; see 50 pLemma 4.24q. We need to bound the terms in (4.11) by the 1 H ńorms over Ωi and Ωj, respectively, of any extension of the values of the function given on the fat face. This require an extension of the finite element face lemma piq to the fat face case. We will therefore consider the Q1´function w which shares piq ij 1 the values of w on the fat face F and which minimizes the H pΩiqńorm among all possible extensions. Thus, we need to bound the zero extension,r to the rest of ij the fat interface, of the function with given values on F and which is Q1´discrete harmonic in Ωi – the part of the interior of Ωi that complements the fat interface – 1 piq by the H pΩiqńorm of w . We will separately estimate the energy contributed i) by the fatr interface and ii) by Ωi. i) The energy contributedr by the fat interface can, after replacing the values of wpiq outside F ij by zero, be estimatedr by twice the energy of wpiq over the entire fat interface plus twice the sum of the squares of the L2ńorms over the thin edges next to the interface between the fat face and its neighboring fatr edges. To see this, we only need to use that

|b|2 ď 2|a ´ b|2 ` 2|a|2, for any pair of 3´vectors, and inspect the expressions on the right-hand side of (2.18). We note that bounds for the L2ńorm over thin edges can be developed by using 50 pLemma 4.16q; see (4.12). A bound with one factor of p1 ` logpH{hqq results. piq piq ii) We also have to compare the energy attributable to Ωi of w and w . We piq note that w vanishes on all but one of the face of Ωi. We first use a standard 1{2 ˜ piq 1 piq trace theorem to bound the H pF qńorm of w by therH pΩiqńormr of w . r r November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

16 L. F. Pavarino, S. Scacchi, O. B. Widlund, S. Zampini

Here, F˜ is a face of the interior cuboid Ωi which is also one of the thin faces of the fat face under consideration. The energy of the discrete harmonic function wpiq in 1{2 ˜ Ωi is then bounded by the square of its rH00 pF q´norm for which we have a bound with a factor Cp1 ` logpH{hqq2, see 50 pLemma 4.26q, and we can conclude that the energyr of the zero extension of wpiq to the rest of the fat face can be bounded by Cp1 ` logpH{hqq2 times the energy of wpiq. We complete the estimate for the fat face contributions by using the Korn inequalities. r Thus, we have proven the following result:

Lemma 4.1 (Fat face lemma).

pkq pkq T pkq pkq pkq 2 pkqT pkq pkq pkq pkq inf pwF ´r q SFF pwF ´r q ď Cp1`logpH{hqq w S w , @w P W . rpkqPRB Here the constant C in independent of H and h but not necessarily of the degree and smoothness of the space of NURBS. Edge contributions. Turning to the fat edges, we will use ideas from the work on the finite element case as developed in 40. We first note that the edge case, in a certain sense, is simpler since a detailed discussion of the minimal energy extension is not necessary. Instead of working with the original bilinear form, we can again 1 50 pLemma 4.19q work with the inner product of H pΩiq times µi and find by using 1 that we can estimate the square of H pΩiqńorm of any function restricted to a fat edge by the number of thin edges of the fat edge times the sum of the squares of the L2ńorms over the thin edges which make up the fat edge. Using 50 pLemma 4.16q, we have

2 2 H θ w 1 C 1 log H h w 1 (4.12) | ip E q|H pΩiq ď p ` p i{ iqq} }H pΩiq

for any thin edge E. Here, w is any Q1 ﬁnite element function and θE the function, which equals 1 at all nodes of the thin edge E and which vanishes at all other nodes, and Hi represents the minimal energy, discrete harmonic extension operator. To provide details, we will now examine the contribution of a fat edge to the norm of PDw. We can estimate the energy contributed by a fat edge by using the pmq pmq principal minors SEE , m “ i, j, k, and `, of the subdomain Schur complements S of the four subdomains that have the fat edge E in common. The square of the norm of the restriction of PDw to E is given by a complicated expression involving the piq sum of squares of the SEE´norm of

piq pjq pkq p`q ´1 pkq pkq piq pSEE ` SEE ` SEE ` SEEq SEEpwE ´ wE q, (4.13)

which is a contribution from Ωi. We will have terms involving pairs of indices i and k of subdomains with a common face and also terms with pairs of subdomains which only have the fat edge E in common. Let us assume the pairs of indices are i, k and j, ` in the latter case. We will only consider the case of i and k in detail. November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

Isogeometric BDDC Deluxe preconditioners for linear elasticity 17

pikq Let us introduce an operator SEE by

p pikq pkq piq pjq pkq p`q ´1 piq piq pjq pkq p`q ´1 pkq SEE :“ SEEpSEE ` SEE ` SEE ` SEEq SEEpSEE ` SEE ` SEE ` SEEq SEE. (4.14) p pkq piq T pikq pkq piq We need a bound for pwE ´ wE q SEE pwE ´ wE q and similar expression for all piq pkq T pkiq piq other pairs of indices with different elements among them pwE ´wE q SEE pwE ´ pkq p wE q. pikq p The operator SEE involves elements of all four Schur complements while those for a subdomain face involves only two. Since the Schur complements generally do not commute, thep algebra and the development of bounds will be more complicated than for the subdomain faces, which only involves two operators. However, since piq piq the SEEńorm of the expression (4.13) can be bounded from above by its SEE ` pjq p`q piq pjq p`q SEE ` SEEńorm, we can, by arguing about two operators, SEE ` SEE ` SEE and pkq pikq pkq SEE, find the upper bound SEE ď SEE. This is one of the two bounds that we pikq piq need; we also need a bound of SEE in terms of SEE. We note that there is a relevantp counter example for the case of three operators in 24 pAppendix Eq which illustratesp that we, in general, cannot expect to derive such pikq piq a bound of SEE in terms of SEE by a simple algebraic argument. We note that this complication is avoided when using the traditional averaging as in 40. However, pkq piq pjq we can returnp to a case only involving the two operators SEE and SEE ` SEE ` pkq p`q pkkq piq SEE ` SEE by working with SEE obtained by replacing the centrally located SEE pkq by SEE in (4.14). By a simple argument using the generalized eigenvalue problem p pkkq pkq defined by these two operators, we obtain the bound SEE ď SEE. pikq pkkq pkq We are now able to combine a bound of SEE from above by SEE and SEE from piq pikq pkkq p above by SEE. By examining SEE and SEE , we see that we can use a bound of piq pkq p p pikq SEE in terms of SEE for the first purpose. We will then obtain our bound for SEE piq p piq p pkq in terms of SEE by showing that SEE is almost equivalent to pµi{µkqSEE. Let MEE be the principal minor of the mass matrix of the degrees of freedomp associated with the mesh given by the knots of E and the Q1 finite element space. We obtain the following bounds, by using 50pLemmas 4.16, 4.19q, with c and C positive constants,

piq piq cSEE ď µiMEE ď Cp1 ` logpHi{hiqqSEE. (4.15)

piq pkq pkq We then obtain SEE ď Cpµi{µkqp1 ` logpH{hqqSEE, and SEE ď Cpµk{µiqp1 ` piq pikq 2 piq logpH{hqqSEE and can conclude that SEE ď Cpp1 ` logpH{hqqq SEE. We now consider the values of the wpmq restricted to an arbitrary thin edge E E of the fat edge E and select a basis for pRB such that r6 vanishes on E. We can now November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

18 L. F. Pavarino, S. Scacchi, O. B. Widlund, S. Zampini

develop a formula, using arguments similar to those that led to (4.10):

5 piq pkq piq piq F ij piq piq E wE ´ wE “ pwE ´ rE q ´ fn pw ´ r qrn n“1 5 ÿ5 F ij pjq pjq E F jk pjq pjq E ` fn pw ´ r qrn ´ fn pw ´ r qrn n“1 n“1 ÿ 5 ÿ F jk pkq pkq E pkq pkq ` fn pw ´ r qrn ´ pwE ´ rE q. n“1 ÿ Here rpiq, rpjq, and rpkq are arbitrary elements of RB and F ij and F jk the two faces on the edge path from Ωi to Ωk. We can now proceed to estimate the Spikqńorm of the different terms in this long formula, the sum of six terms. The 3 piq piq 2 first term can be estimated by Cp1 ` logpH{hqq µ }w ´ r } 1 , and the sixth i H pΩiq p pkq pkq 2 by Cp1 ` logpH{hqqµ }w ´ r } 1 . The second term can also be estimated k H pΩkq 3 piq piq 1 by Cp1 ` logpH{hqq µi}w ´ r }H pΩiq by using (4.5) and an estimate of the E 50 pLemma 4.17q energy of the rn obtained by using . The fifth term can be estimated in the same way. Finally, the third and fourth terms can be estimated by Cp1 ` 3 pjq pjq 2 logpH{hqq TOL ‹ µ }w ´ r } 1 ; here we use the best of the two bounds j H pΩj q pikq pjq piq pkq available for SEE and a bound of SEE from above by SEE or SEE as well as Definition 4.1. Given thatp the thin edge E can be chosen as any of the thin edges of E, we have completed what is needed for the fat edges in the quasi-monotone case. Vertex contributions. The estimates of the contributions of the fat vertices can be estimated in very much the same way as in 40. The operators replacing those of (4.14) are even more complicated but given that we are now working on subspaces which are of fixed and small dimensions, they can easily be bounded from above in terms of µi. The sums of five terms used when estimating the edge contributions, will now be replaced by just three. For any knot V of the fat vertex, F F F 40pSubsec. 8.4.3q we will use a basis for RB for which r4 , r5 , and r6 vanish at V; cf. . 3 F F At V, we then have r “ n“1 fn prqrn for any of the faces that we pass through when following a vertex path. We note that the vertex path now will involve passing ř through three subdomain faces when we pass between subdomains that only have a vertex in common. The estimates are worked out for one knot V of V at a time. The energy of a piq piq term of the form pw pVq ´ r pVqqϕV , where ϕV is the Q1ńodal basis function piq piq 2 for knot V, is bounded by C}w ´ r } 1 since the energy of ϕ is of order h H pΩiq V and for any element K

2 C 2 }w} 8 ď }w} 1 ; (4.16) L pKq h H pKq see 50 pLemma 4.16q. When estimating the other terms corresponding to those in (4.10), we can again take advantage of the small energy of the nodal basis function November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

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and we can therefore be more lenient in terms of the jumps of the values of µ; see 40 pSubsec. 8.4.3q and Deﬁnition 4.2. Thus, the second term in the formula that replace 3 F ij piq (4.10) is ´ n“1 fn pw ´ r qrnpVqϕV , which can be estimated by Cphi{Hiqp1 ` piq piq logpH{hqq}w ´ r } 1 . The bound of all the other terms can otherwise be ř H pΩiq developed in a way very close to what was done for the fat subdomain edges.

4.3. The general case We now turn to the case when we do not necessarily have quasi-monotone coefficients. An example of such a case is the 3D checker board distribution of the µi. For this case, moving around each subdomain edge, we find that the Laméparam- eter µ can alternate between a large value and a value that is much smaller. We note that if quasi-monotonicity fails for just a few subdomain edges, we can modify the arguments of the previous subsection by extending the path between pairs of subdomains that only have an edge in common to work with paths through some other nearby subdomains and obtain the same strong result as before by using the same set of primal constraints. By choosing additional primal constraints when analyzing the contributions from the fat edges and vertices, we can establish the following result. We note that the additional primal constraints are only needed for fat edges and vertices for which Definitions 4.1 and 4.2 do not hold. L

Theorem 4.2. Assume that for each fat edge E, that does not satisfy Deﬁnition 4.1, there is a thin edge E in the center of E, which is fully primal, i.e., there are ﬁve primal constraints associated with the thin edge as given by (4.1) for 1 ď k ď 5 and an additional primal constraint given by a volume integral, dx :“ dx1dx2dx3, over the fat edge E:

w ¨ r6dx gEpwq :“ E . (4.17) 6 r ¨ r dx şE 6 6 We can also handle cases when thereş is no thin edge at the center of the fat edge Es; in such a case, we can simply introduce an auxiliary thin edge at the center of E and use it when deﬁning the six primal constraints. Assume further that for each fat vertex which does not satisfy Deﬁnition 4.2, there are three primal constraints which makes the three components of the elements of WΓ continuous at one knot V, centered in the fat vertex V and three additional primal constraints given by volume integrals, dx :“ dx1dx2dx3, over the fat vertex V :Ă

w ¨ rkdx gV :“ V for k “ 4, 5, 6. (4.18) k r ¨ r dx şV k k In case there is no knot atş the center of V , we can again instead work with a point at the center of the fat vertex. November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

20 L. F. Pavarino, S. Scacchi, O. B. Widlund, S. Zampini

Under these assumptions, the condition number of the BDDC deluxe algorithm satisﬁes

´1 2 κpM Aq ď }ED}S ď Cp1 ` logpH{hqq ,

with C independent of the number of subdomains,q the subdomains diameters Hi, the mesh sizes hi, and the elastic coeﬃcients but can grow with the number of knots across the fat interface.

Proof. In our proof, we will again separately bound the face, edge, and vertex contributions to the norm of PDw. Face contributions The analysis related to the subdomain faces requires noth- ing new since each of these faces is fully primal, i.e., is associated with a set of six primal constraints which satisfies the bounds of (4.5). This is guaranteed by the fact that we enrich the set of primal constraints previously in use. Edge contributions We note that in the previous subsection, we were able to use primal constraints related to different subdomain edges to handle terms related to the different faces of the subdomains through which our edge paths pass. Here, we instead will use, in our arguments, only primal constraints that are directly related to the individual subdomain edges under consideration and no longer argue about edge paths. We start by selecting one thin edge centered in the fat subdomain edge and turning it into a fully primal edge by adopting the five first functionals given E by (4.1); we recall that g6 serves no good purpose but that we were still able to establish (4.2) at the center of the fat edge but only for x P E, i.e., for just one of the thin edges of the fat edge. These five first constraints will work well for that single thin edge but we need at least one additional, sixth, linear functional to handle the other thin edges of the same fat edge. We could make them all fully primal but we are anxious to develop a small coarse space. This is the reason for introducing E E the constraint g6 which satisfies g6 priq “ δi6. We need a bound for this new linear functional but we do not need a bound as strong as (4.4) since r6 vanishes on the thin edge E and that the other thin edges are within a multiple of the mesh size h from that thin edge; the values of r6 are therefore small for all x P E. When estimating the energy contributed by an arbitrary thin edge of the fat subdomain edge, we will consider the expression 5 E E E E gi pwqri ` g6 pwqr6 . i“1 ÿ Each of the first five terms can be estimated using (4.4). The sixth can also be bounded well enough: 2 2 }w} 2 p1 ` logpH{hqq}w} 1 E 2 L pEq 2 H pΩiq |g6 pwq| ď 2 ď CH , }r6}L2pEq |E| where |E| is the volume of E and C a constant. We can now take advantage of the fact that r6 has small values on all of the thin edges of E and we can show, by November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

Isogeometric BDDC Deluxe preconditioners for linear elasticity 21

elementary means, that for any of the thin edges E1 of E

2 C|E| |HipθE1 r6q| 1 ď . H pΩiq H2 When developing our bound, we will consider expressions given by

5 piq piq E piq piq E piq piq w ´ r ´ p gi pw ´ r qri ` g6 pw ´ r qr6q i“1 ÿ for an arbitrary rpiq P RB. We can now complete our argument by referring to 16 pSec. 3q where it is shown that the square of the norm of the restriction of PDw to E, for the deluxe variant of BDDC, can be estimated by piqT piq piq pjqT pjq pjq pkqT pkq pkq p`qT p`q p`q 4pwE,∆SEEwE,∆ ` wE,∆ SEEwE,∆ ` wE,∆ SEEwE,∆ ` wE,∆ SEEwE,∆q. (4.19)

piq piq piq pjq pjq pjq Here wE,∆ :“ wE ´ r , wE,∆ :“ wE ´ r , etc. Vertex contributions We can use similar arguments for the contributions from a fat vertex. In particular, we can use a bound quite similar to (4.19). We choose the origin of a coordinate system at the centrally located knot V and consider the expression

3 6 V w ¨ ekrk ` gk pwqrk. k“1 k“4 ÿ ÿ We can focus on the last three terms in this sum. By using elementary inequalities and (4.16), we ﬁnd that 2 2 }w} 2 }w} 1 gV w 2 L pV q CH2 H pΩiq . | k p q| ď 2 ď 2{3 }rk}L2pV q h|V |

2 2{3 2 We also use that }ϕ r } 1 ď Ch|V | {H , where ϕ is a nodal basis function, V k H pΩiq V to complete the necessary bound. The rest of the proof can be closely modeled on the arguments given in the previous subsection.

We note that in the study of the FETI–DP algorithm for ﬁnite element approximations of elasticity, 40, additional primal constraints were also introduced and that experimental work reported in 38 demonstrated that the use of a smaller set of primal constraints, such as that of Subsection 4.3, led to a poor performance. Similar results are reported at the end of Section 5 of this paper.

5. Numerical results All the experiments reported are for problems in three dimensions. The elasticity system is discretized on a cubic domain and on a twisted pipe section, a quite distorted image of a cube, see Fig. 2, using isogeometric NURBS spaces with a November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

22 L. F. Pavarino, S. Scacchi, O. B. Widlund, S. Zampini

a) cubic domain b) twisted pipe section

Fig. 2. Computational domains used in the numerical tests. Colors denote the value of the ﬁrst displacement component.

uniform mesh size h, polynomial degree p, and regularity k. The domain is decomposed into K non-overlapping subdomains of characteristic size H. The discrete problems are solved by the preconditoned conjugate gradient, (PCG), method with the isogeometric BDDC deluxe preconditioners, with a zero initial guess and a stopping criterion of a 10´6 reduction of the Euclidean norm of the PCG residual. In our tests, we study how the convergence rate of the BDDC preconditioner depends on h, K, p, k, and on the jumps in the coeﬃcient of the elliptic problem. In all tests, the BDDC condition number is essentially the maximum eigenvalue of the preconditioned operator, since its minimum eigenvalue is always very close to 1. The 3D parallel tests have been performed using the PetIGA library 21 and the PETSc library 3 with its PCBDDC preconditioner, contributed to the PETSc library by Stefano Zampini, see 53, and run on the parallel machines Marconi- A1 of Cineca (http://www.hpc.cineca.it/hardware/marconi) and Shaheen-XC40 of KAUST (https://www.hpc.kaust.edu.sa/content/shaheen-ii).

5.1. Conditioning of the unpreconditioned isogeometric elasticity system In Fig. 3 we report on the conditioning of the unpreconditioned isogeometric elasticity system as a function of the inverse of the mesh size, 1{h, for a fixed p “ 3, k “ 2 (left column) and of the polynomial degree p, with maximal regularity k “ p ´ 1 and a fixed 1{h “ 32 (right column). We report both the CG iteration counts (top row) and the condition number of the unpreconditioned elasticity operator, computed during the CG iterations using the connection of the CG algorithm to Lanczos’ eigenvalue method. The results show that the condition number of the isogeometric 3D elasticity stiffness matrix grows, as expected, as Oph´2q, when h-refinement is performed, while it grows as Opp2q when k-refinement is performed. We observe that, already in the case of the smallest p “ 2, the condition number of the unpreconditioned system is very large and on the order of 104. November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

Isogeometric BDDC Deluxe preconditioners for linear elasticity 23

cubic domain twisted pipe section 3 3 10 10 O(p) O(h-1)

102 102 1/2

-1/2 O(p ) ITERATIONS O(h ) ITERATIONS

101 101 12 18 24 30 36 42 48 2 3 4 5 6 1/h POLYNOMIAL DEGREE p

106 105 3

m -3 O(p )

O(h ) m /

/ 5

M 10 M

104 4 -2 10 O(p2)

COND. = = COND. O(h ) COND. = = COND.

3 10 103 12 18 24 30 36 42 48 2 3 4 5 6 1/h POLYNOMIAL DEGREE p

Fig. 3. Conditioning of the unpreconditioned isogeometric elasticity system, with constant Lamé parameters, as a function of 1{h, the inverse of the mesh size, for fixed p “ 3, k “ 2 (left column) and polynomial degree p and k “ p ´ 1 for fixed 1{h “ 32 (right column). Top row: CG iteration counts. Bottom row: condition number of the unpreconditioned elasticity operator.

5.2. Primal spaces for BDDC deluxe We then investigate the performance of our isogeometric BDDC deluxe preconditioner for a number of choices of primal spaces of increasing dimension. We start with several reduced primal spaces with only a few primal degrees of freedom (dofs) for individual equivalence classes. When considering constraints arising from rigid body modes, we restrict their respresentative vectors in the discrete space to each relevant equivalence class and run a Singular Value Decomposition (SVD) on such subvectors, in order to eliminate any possible null rotation or linear dependency.

‚ E3: use only 3 primal dofs associated to the edge averages of the three displacement components on one thin edge E for each fat edge E. This is the smallest primal space that we have considered and it is the one analyzed in Theorem 4.1. November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

24 L. F. Pavarino, S. Scacchi, O. B. Widlund, S. Zampini

‚ V9E6: use 3 primal dofs per fat vertex V (displacement components in the middle) + SVD of rigid body modes on the remaining dofs of the fat vertex V (6 primal dofs) + SVD of rigid body modes on each fat edge E (6 primal dofs). ‚ V3E6: use 3 primal dofs per fat vertex V (displacement components in the middle) + SVD of rigid body modes on the thin edge E in the middle of each fat edges E (5 constraints per edge for the cube case, 6 for the twisted pipe case). ‚ V6E6: use 6 primal dofs per fat vertex V (3 displacement components in the middle and 3 volume averages) + SVD of rigid body modes on the thin edge E in the middle of each fat edges E (5 constraints per edge for the cube case, 6 for the twisted pipe case). This choice corresponds to the case covered by our theory (Th. 4.2). ‚ V9E6F6:V9E6 + SVD of rigid body modes on each fat face (6 primal dofs). ‚ V3E6F6:V3E6 + SVD of rigid body modes on the thin face F in the middle of each fat face F (6 primal dofs). We also consider rich (expensive) primal spaces that include all the p3 dofs of each fat vertex V , the averages on all p2 thin edges E of each fat edge E and the averages on all p thin faces F of each fat face F ; in all cases k “ p ´ 1 :

‚ V = V3p3 : all displacement components for each fat vertex are made primal; ‚ VE = V3p3 E3p2 :V3p3 + edge averages (one per displacement component) for all thin edges E of each fat edge E;

‚ VEF = V3p3 E3p2 F3p:V3p3 E3p2 + face averages (one per displacement component) for all thin faces F of each fat face F .

5.3. Weak scalability test We perform a weak scalability test for the unit cube and for the twisted pipe section domain. The number of subdomains K is increased from 23 to 83, keeping the local size H{h “ 6, the spline polynomial degree p “ 3 and regularity k “ 2 ﬁxed. The results reported in Fig. 4 show that BDDC with all the three rich coarse spaces are scalable, since the condition numbers and iteration counts seem to ap- proach constant values when increasing the number of subdomains. As expected, larger primal spaces yield fewer iterations. In particular, adding the edge averages to the vertex constraints in the VE primal space leads to a large improvement over V, while adding further the face averages in the VEF primal space gives only a marginal improvement over VE. For the twisted pipe section domain, all the three methods seem to be quite sensitive to the domain deformations, in particular for the V primal space, but the VE and VEF coarse spaces outperform V by an order of magnitude in terms of condition number. Moreover, the results in Fig. 4 show that the reduced coarse spaces are also scalable, in both the cube test (top plots) and twisted pipe test (bottom plots). Dif- November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

Isogeometric BDDC Deluxe preconditioners for linear elasticity 25

cubic domain twisted pipe section 100 400

80 300

60 200 40

ITERATIONS 100 20 E 1 VE 0 0 0 8 64 125 216 343 512 8 64 125 216 343 512 VE 1 SUBDOMAINS SUBDOMAINS K VEF 0 VEF 300 1 V 3000 VE VEF 200

2000

COND.

M 100 1000

0 0 8 64 125 216 343 512 8 64 125 216 343 512 SUBDOMAINS K SUBDOMAINS K

Fig. 4. BDDC deluxe weak scalability. Left column: cubic domain. Right column: twisted pipe section. Top row: BDDC iteration counts. Bottom row: maximum eigenvalue λM « condition number of the BDDC preconditioned operator.

ferently from the rich coarse spaces, the addition of face primal constraints (V9E6F6 and V3E6F6) now deﬁnitely improves the BDDC performance and the degradation in the twisted pipe test then seems less severe.

5.4. Quasi-optimality test Fig. 5 reports the results of a quasi-optimality test, with increasing values of H{h “ 5, 6, ¨ ¨ ¨ , 12 for fixed spline parameters p “ 3, k “ 2 and number of subdomains K “ 3 ˆ 3 ˆ 3. Since the domain and its subdivision are fixed, we are here varying the mesh size h. First, we consider the rich primal spaces V, VE, and VEF. For the unit cube, the results show a linear dependence of the condition number on H{h for the primal space V. The addition of edge averages (VE) seems sufficient to obtain a quasi- optimal method, with a logarithmic growth of the condition number as a function of H{h. As before, VEF yields only a marginal improvement over VE. The strong deformation of the twisted pipe test, leads again to much worse condition numbers and CG iteration counts in comparison to the unit cube test, but the quasi-optimal November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

26 L. F. Pavarino, S. Scacchi, O. B. Widlund, S. Zampini

cubic domain twisted pipe section 80 250

200 60

150

40 100

ITERATIONS 20 50 E 1 VE 0 0 0 VE 4 6 8 10 12 4 6 8 10 12 1 VEF H/h H/h 0 VEF 1 200 V 2500 VE VEF 150 2000

1500

COND. 100

1000 M 50 500

0 0 4 6 8 10 12 4 6 8 10 12 H/h H/h

Fig. 5. BDDC deluxe quasi-optimality with respect to H{h . Left column: cubic domain. Right column: twisted pipe section. Top row: BDDC iteration counts. Bottom row: maximum eigenvalue λM « condition number of the BDDC preconditioned operator.

behavior of the BDDC with VE and VEF coarse spaces is conﬁrmed. Focusing now on the reduced coarse spaces, Fig. 5 shows a better than expected BDDC performance, since the plots of iteration counts and maximum eigenvalues, even if irregular, seem not to increase with H{h (except VE0 in the cube test). We do not have an explanation for this unexpected behavior.

5.5. Dependence on p In this test, we study the BDDC deluxe convergence rate for increasing polynomial degree p “ 2, 3, ..., 6 and maximal regularity k “ p ´ 1. The domains considered are the unit cube and the twisted pipe section, discretized with the mesh size h “ 1{32 and subdivided into K “ 4 ˆ 4 ˆ 4 subdomains. As in the previous tests, we ﬁrst consider the three rich coarse spaces. From the results reported in Fig. 6, in case of the unit cube domain, we cannot recognize a clear behavior for V. However, for VE and VEF both CG iterations and condition numbers are initially quite small, but for p “ 6 they start to increase signiﬁcantly. Again, on the deformed domain of the twisted pipe test, the BDDC solver perfor- November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

Isogeometric BDDC Deluxe preconditioners for linear elasticity 27

cubic domain twisted pipe section 103 103

102

101 ITERATIONS

E 1 0 1 VE 10 10 0 2 3 4 5 6 2 3 4 5 6 VE 1 POLYNOMIAL DEGREE p POLYNOMIAL DEGREE p VEF 0 VEF 105 107 1 V VE VEF

103 4

COND. 10

100 101 2 3 4 5 6 2 3 4 5 6 POLYNOMIAL DEGREE p POLYNOMIAL DEGREE p

Fig. 6. BDDC deluxe dependence on ps. Left column: cubic domain. Right column: twisted pipe section. Top row: BDDC iteration counts. Bottom row: maximum eigenvalue λM « condition number of the BDDC preconditioned operator.

mance degrades considerably in comparison to the unit cube test, particularly for the V coarse space, while VE and VEF seem to be more robust and with nearly the same performance. Surprisingly, in the twisted pipe test, CG iterations and condition numbers decrease when increasing p for all the three coarse spaces considered. On the other hand, Fig. 6 shows that BDDC with reduced coarse spaces has a stronger dependence on the polynomial degree p and we could run tests only up to p “ 4 (p “ 3 for V9E6 and V9E6F6 in the twisted pipe case). The addition of face primal constraints helps the BDDC performance, particularly in the cube tests, but the iterations and maximum eigenvalue seem to grow more than linearly with p.

5.6. Robustness with respect to jumping coeﬃcients Finally, we investigate the robustness of our BDDC solver with respect to jumping material parameters. The unit cube is subdivided into K “ 27 “ 3 ˆ 3 ˆ 3 subdomains, ordered lexicographically plane by plane as illustrated in the left plot of Table 1. Denote by rank “ 0, ..., 26 the index of the generic subdomain. The spline parameters are ﬁxed at p “ 3 and k “ 2, and the ratio H{h “ 6. With these November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

28 L. F. Pavarino, S. Scacchi, O. B. Widlund, S. Zampini

E3 V3E6 V3E6F6 J nit λM nit λM nit λM quasi-monotone: 10J if modprank, 3q “ 0 or 2, rank: cf “ 10´J if modprank, 3q “ 1 " 0 53 1.22e+2 38 8.84e+1 21 1.81e+1 1 48 1.37e+2 35 1.00e+2 23 3.49e+1 2 42 1.37e+2 33 1.02e+2 19 3.69e+1 3 40 1.37e+2 33 1.02e+2 18 3.70e+1 4 35 1.37e+2 28 1.02e+2 14 3.66e+1 checkerboard: 10J if modprank, 2q “ 0 c “ f 10´J if modprank, 2q “ 1, ν “ 0.3 " E “ 210 ¨ cf 0 53 1.22e+2 38 8.84e+1 21 1.82e+1 1 41 6.53e+1 29 4.85e+1 24 3.65e+1 2 38 6.25e+1 25 4.58e+1 25 4.56e+1 3 38 6.25e+1 25 4.58e+1 25 4.58e+1 4 38 6.25e+1 25 4.58e+1 25 4.58e+1

Table 1. Quasi-monotone and checkerboard jumping coeﬃcients: iteration counts nit and maximum eigenvalue λM (« condition number) as functions of the jumping parameter J for ﬁxed H{h “ 6, K “ 3 ˆ 3 ˆ 3, and spline parameters p “ 3, k “ 2.

choices, the total number of dofs is 41472, with that on the interface being 23976. The Poisson ratio is ﬁxed to 0.3 and the Young modulus is E “ 210 ¨ cf , with the coeﬃcient cf varying as follows:

10J if mod prank, 3q “ 0 or 2 ‚ quasi-monotone test: c “ f 10´J if mod prank, 3q “ 1, " with J “ 0, 1, 2, 3, 4; 10J if mod prank, 2q “ 0 ‚ checkerboard test: c “ f 10´J if mod prank, 2q “ 1, " with J “ 0, 1, 2, 3, 4.

In addition to the coarse spaces V3E6 and V3E6F6 described previously, we consider here also the coarse space introduced in Theorem 4.1, consisting of 3 primal dofs per fat edge and denoted by E3. The size of the coarse space is 180, 300 and 624 for the E3,V3E6 and V3E6F6 cases, respectively. The results reported in Table 1 show that all three methods are robust in both the quasi-monotone and checkerboard cases. We finally consider a harder test with nonquasi-monotone coefficients considered in 38 (Table 5 and Fig. 9). On a cubic domain decomposed into 3ˆ3ˆ3 subdomains as before, we now choose the ratio H{h “ 7. As before, the Poisson ratio is fixed November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

Isogeometric BDDC Deluxe preconditioners for linear elasticity 29

nonquasi-monotone: 10J in subdomains 0, 2, 6, 8, 13, 18, 20, 24, 26 c “ f 1 otherwise " p “ 1, k “ 0 p “ 2, k “ 1 p “ 3, k “ 2 J nit λM nit λM nit λM E3 primal space 0 12 4.78e+0 50 8.87e+1 137 1.92e+3 1 15 9.79e+0 50 8.12e+1 160 1.91e+3 2 16 1.38e+1 49 9.19e+1 163 3.66e+3 3 22 8.44e+1 65 4.65e+2 203 1.85e+4 4 26 7.79e+2 96 4.21e+3 301 1.65e+5 5 34 7.72e+3 132 4.15e+4 463 1.63e+6 6 44 7.72e+4 183 4.14e+5 ą 500 1.63e+7 V6E6 primal space 0 10 2.86 20 1.03e+1 52 8.22e+1 1 12 4.60 21 1.19e+1 57 8.98e+1 2 10 4.34 20 1.33e+1 62 1.31e+2 3 9 4.28 20 1.79e+1 65 2.23e+2 4 9 4.28 20 2.26e+1 64 2.99e+2 5 8 4.28 20 2.37e+1 60 3.15e+2 6 8 4.28 20 2.39e+1 59 3.17e+2

Table 2. Nonquasi-monotone jumping coeﬃcients: iteration counts nit and maximum eigenvalue λM (« condition number) as functions of the jumping parameter J for ﬁxed H{h “ 7, K “ 3ˆ3ˆ3.

to 0.3 and the Young modulus is E “ 210 ¨ cf , but now the coeﬃcient cf varies as follows:

‚ nonquasi-monotone test: 10J in subdomains 0, 2, 6, 8, 13, 18, 20, 24, 26 c “ f 1 otherwise " with J “ 0, 1, 2, 3, 4, 5, 6;

The results reported in Table 2 for splines with degree p “ 1, 2, 3 show, as suggested by our main Theorems 4.1 and 4.2, that the smaller primal space E3 is not robust with respect to the magnitude J of the coeﬃcient jumps, while the primal space V6E6 analyzed in Th. 4.2 is robust. For both primal spaces, the BDDC performance worsen with increasing degree p, as shown already in the easier constant coeﬃcient tests of Fig. 6, but the BDDC iterations and condition number growth is contained (and robust in J) for the V6E6 primal space, while the growth for the thin E3 primal space is catastrophic when J increases, already for p “ 2 and 3. November 20, 2017 14:43 WSPC/INSTRUCTION FILE TR2017-988

30 L. F. Pavarino, S. Scacchi, O. B. Widlund, S. Zampini

6. Conclusions We have developed and analyzed BDDC deluxe preconditioners for isogeometric analysis discretizations of three-dimensional compressible linear elasticity. We have proved that the BDDC deluxe preconditioners, with small coarse spaces, are scalable and quasi-optimal. Parallel 3D numerical tests have validated the theoretical results, not only for the small coarse spaces covered by the theory but also for other choices. We have also studied numerically the performance of the preconditioners for increasing degrees and regularity of the splines. Future work will be devoted to the adaptive selection of the primal constraints for BDDC deluxe and to the extension of dual-primal preconditioners to isogeometric discretizations of almost incompressible linear elasticity in mixed form and of Stokes problems.

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