Domain Decomposition Algorithms for Saddle Point Problems

In this paper, we introduce some domain decomposition methods for saddle point problems with or without a penalty term, such as the Stokes system and the mixed formulation of linear elasticity. We also consider more general nonsymmetric problems, such as the Oseen system, which are no longer saddle point problems but can be studied in the same abstract framework which we adopt. Several approaches have been proposed in the past for the iterative solution of saddle point problems. We recall here: Uzawa’s algorithm and its variants (Arrow, Hurwicz, and Uzawa [1], Elman and Golub [24], Bramble, Pasciak, and Vassilev [10], Maday, Meiron, Patera, and Rønquist [38]); multigrid methods (Verfürth [54], Wittum [55], Braess and Blömer [7], Brenner [11]); preconditioned conjugate gradient methods for a positive definite equivalent problem (Bramble and Pasciak [8]); block–diagonal preconditioners (Rusten and Winther [50], Silvester and Wathen [51], Klawonn [31]); block–triangular preconditioners (Elman and Silvester [25], Elman [23], Klawonn [32], Klawonn and Starke [34], Pavarino [43]). Some of these approaches allow the use of domain decomposition techniques on particular subproblems, such as the inexact blocks in a block preconditioner. In this paper, we propose some alternative approaches based on the application of domain decomposition techniques to the whole saddle point problem, discretized with either h-version finite elements or spectral elements. We will consider both a) overlapping Schwarz methods and b) iterative substructuring methods. We refer to Smith, Bjørstad, and Gropp [52] or Chan and Mathew [18] for a general introduction to domain decomposition methods.