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Theoretically Supported Scalable FETI for Numerical Solution of Variational Inequalities

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Theoretically Supported Scalable FETI for Numerical Solution of Variational Inequalities Theoretically supported scalable FETI for numerical solution of variational inequalities Zden·ekDost¶aland David Hor¶ak ¤ Abstract The FETI method with a natural coarse grid is combined with recently proposed optimal algorithms for the solution of bound and/or equality con- strained quadratic programming problems in order to develop a scalable solver for elliptic boundary variational inequalities such as those describing equilib- rium of a system of bodies in mutual contact. A discretized model problem is ¯rst reduced by the duality theory of convex optimization to the quadratic programming problem with bound and equality constraints. The latter is then modi¯ed by means of orthogonal projectors to the natural coarse grid intro- duced by Farhat, Mandel and Roux. Finally the classical results on linear scalability for linear problems are extended to boundary variational inequal- ities. The results are validated by numerical experiments. The experiments also con¯rm that the algorithm enjoys the same parallel scalability as its linear counterpart. Keywords: Domain decomposition, variational inequality, scalability, parallel algorithms, FETI Acknowledgment: This research is supported by grant No. 101/04/1145 of the Grant Agency of the Czech Republic and by projects of the Ministry of Education No. 1ET400300415 and ME641. 1 Introduction The FETI (Finite Element Tearing and Interconnecting) domain decomposition method was originally proposed by Farhat and Roux [26] for parallel solving of the linear problems described by elliptic partial di®erential equations. Its key ingredient ¤FEI VSB-Technical· University of Ostrava, CZ-70833 Ostrava, Czech Republic 1 is decomposition of the spatial domain into non-overlapping subdomains that are "glued" by Lagrange multipliers, so that, after eliminating the primal variables, the original problem is reduced to a small, relatively well conditioned, typically equality constrained quadratic programming problem that is solved iteratively. The time that is necessary for both the elimination and iterations can be reduced nearly proportion- ally to the number of the processors, so that the algorithm enjoys parallel scalability. Observing that the equality constraints may be used to de¯ne so called "natural coarse grid", Farhat, Mandel and Roux [25] modi¯ed the basic FETI algorithm so that they were able to adapt the results by Bramble, Pasciak and Schatz [5] to prove its numerical scalability, i.e. asymptotically linear complexity. The comprehensive review of the mathematical results related to the FETI methods may be found in the monograph by Tosseli and Widlund [38]. If the FETI procedure is applied to an elliptic variational inequality, the resulting quadratic programming problem has not only the equality constraints, but also the non-negativity constraints. Even though the latter is a considerable complication as compared with linear problems, it seems that the FETI procedure should be even more powerful for the solution of variational inequalities than for the linear problems. The reason is that FETI not only reduces the original problem to a smaller and better conditioned one, but it also replaces for free all the inequalities by the bound constraints. Promising experimental results by Dureisseix and Farhat [22] supported this claim and even indicated numerical scalability of their method. Similar results were achieved also for the FETI{DP (Dual{Primal) method introduced by Farhat et al. [24]. The FETI{DP method is very similar to the original FETI, the only di®erence is that it enforces the continuity of displacements at corners on primal level. A new Lagrange multipliers algorithm, FETI{C, based on FETI{DP and on active set strategies with additional planning steps and preconditioning, was introduced by Farhat et al. [1, 22]. Its scalability was demonstrated experimentally. Another approach yielding an experimental evidence of scalability was proposed by Dost¶al,Friedlander, Gomes, Santos and Hor¶ak[12, 13, 14]. The algorithm com- bined FETI with a special variant of the augmented Lagrangian method [10]. Scal- ability was later proved for an algorithm that enforced the equality constraints by the optimal dual penalty [15, 16] and solved the resulting bound constrained prob- lem by recent in a sense optimal algorithms [7, 21]. Using the same algorithms, Dost¶al,Hor¶akand Stefanica then proved numerical scalability for a FETI{DP algo- rithm applied to the coercive problems discretized by means of either nodal [18] or mortar [19] Lagrange multipliers. Most recently, the scalability results were proved also for semicoercive problems [20]. The rate of convergence was given in terms of the e®ective condition number of the dual Schur complement of the sti®ness matrix which was proved to be bounded by CH2=h2, where C is a constant independent of the discretization and decomposition parameters h and H, respectively. The es- timates did not assume any preconditioning. Indeed, numerical experiments by the present authors, V. Vondr¶akand M. Lesoinne indicated that the performance of our FETI{DP based algorithms may be considerably improved by preconditioning. 2 It should be noted that the e®ort to develop scalable solvers for variational in- equalities was not restricted to FETI. For example, using ideas related to Mandel [34], Kornhuber, Krause, Sander and Wohlmuth [30, 31, 40, 32, 33] gave an experimen- tal evidence of numerical scalability of the algorithm based on monotone multigrid. Probably the ¯rst theoretical results concerning development of scalable algorithms were proved by SchÄoberl [36, 37]. In this paper, we use the FETI method with a natural coarse grid to develop a scalable algorithm for numerical solution of both coercive and semicoercive varia- tional inequalities. The rate of convergence is again given in terms of the e®ective condition number of the dual Schur complement of the sti®ness matrix, but this time it is bounded by CH=h. The paper is organized as follows. After describing a model problem, we briefly review the FETI methodology [12] that turns the variational inequality into the well conditioned quadratic programming problem with bound and equality constraints. Then we review our algorithms for solution of the resulting bound and equality constrained quadratic programming problem whose rate of convergence may be ex- pressed in terms of bounds on the spectrum of the dual Schur complement matrix [21, 8, 9]. Finally we present the main results about optimality of our method and give results of numerical experiments with parallel implementation of the algorithm in PETSc [3]. 2 Model problem For the sake of simplicity, we shall reduce our analysis to a simple model problem, but our reasoning is valid also in more general cases, including contact problems of 2D and 3D elasticity, provided that the conditions exploited in the proof of the results by Farhat, Mandel and Roux [25] are satis¯ed. Let ­ = ­1 [ ­2; ­1 = (0; 1) £ (0; 1) and ­2 = (1; 2) £ (0; 1) denote open domains with boundaries ¡1, 2 i i i i ¡ and their parts ¡u,¡f ,¡c formed by the sides of ­ ; i = 1; 2 as in Figure 1a or Figure 1b. Let H1(­i); i = 1; 2 denote the Sobolev space of the ¯rst order in the space L2(­i) of the functions on ­i whose squares are integrable in the sense of Lebesgue. Let i © i 1 i i i ª V = v 2 H (­ ): v = 0 on ¡u denote the closed subspaces of H1(­i); i = 1; 2, and let 1 2 © 1 2 2 1 ª V = V £ V and K = (v ; v ) 2 V : v ¡ v ¸ 0 on ¡c denote the closed subspace and the closed convex subset of H = H1(­1) £ H1(­2), respectively. The relations on the boundaries are in terms of traces. On H we shall 3 de¯ne a symmetric bilinear form Z µ ¶ X2 @ui @vi @ui @vi a(u; v) = + d­ i @x @x @y @y i=1 ­ and a linear form X2 Z `(v) = f ivid­; i i=1 ­ where f i 2 L2(­i); i = 1; 2 are the restrictions of 8 < ¡1 for (x; y) 2 (0; 1) £ [0:75; 1) f(x; y) = 0 for (x; y) 2 (0; 1) £ [0; 0:75) and (x; y) 2 (1; 2) £ [0:25; 1) : ¡3 for (x; y) 2 (1; 2) £ [0; 0:25) for coercive problem and 8 < ¡3 for (x; y) 2 (0; 1) £ [0:75; 1) f(x; y) = 0 for (x; y) 2 (0; 1) £ [0; 0:75) and (x; y) 2 (1; 2) £ [0:25; 1) : ¡1 for (x; y) 2 (1; 2) £ [0; 0:25) for semicoercive problem. Thus we can de¯ne a problem to ¯nd 1 min q(u) = a(u; u) ¡ `(u) subject to u 2 K: (2.1) 2 −1 f -3 f −3 -1 Ω 2 Ω 2 0.25 0.75 0.25 0.75 0.25 0.25 0.75 0.75 Ω 1 Ω 1 1 1 1 1 ΓΓΓΓ1 1 2 Γ 2 ΓΓΓΓ1 1 2 uf c f u uf c f Fig. 1a: Coercive model problem Fig. 1b: Semicoercive model problem Figure 1: Model problems We shall consider two variants of the Dirichlet data. In the ¯rst case, both the membranes are ¯xed on the outer edges as in Figure 1a, so that 1 2 2 2 ¡u = f(0; y) 2 IR : y 2 [0; 1]g; ¡u = f(2; y) 2 IR : y 2 [0; 1]g: 4 i Since the Dirichlet conditions are prescribed on parts ¡u; i = 1; 2 of the boundaries of the both membranes with positive measure, the quadratic form a is coercive which guarantees both existence and uniqueness of the solution [28, 27]. In the second case, only the left membrane is ¯xed on the outer edge and the right membrane has no prescribed displacement as in Figure 1b, so that 1 2 2 ¡u = f(0; y) 2 IR : y 2 [0; 1]g; ¡u = ;: Even though a is in this case only semide¯nite, the form q is still coercive due to the choice of f so that it has again the unique solution [28, 27].
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