Istituto di Matematica Applicata e Tecnologie Informatiche     
Pavarino L. F. Overlapping Schwarz methods for elasticity and Stokes problems. Preprint ercim.cnr.ian//1997-1030, 1997.
We introduce and study a class of overlapping domain decomposition preconditioners for saddle point problems with a penalty term, such as Stokes problems and the mixed formulations of the linear elasticity system. Finite element and finite difference discretizations of these problems produce large symmetric indefinite linear systems. We solve these indefinite systems by using an appropriate accelerator, such as GMRES, with an indefinite preconditioner obtained by overlapping Schwarz techniques. This preconditioner is based on the solution of local saddle point problems on overlapping subdomains and the solution of a coarse saddle point problem. As in the positive definite case, our method is parallelizable, scalable and has a simple coarse problem. The numerical experiments reported in the last section indicate that the rate of convergence of the preconditioned operator is independent of the mesh size $h$, the number of subdomains $N$ and the penalty parameter. We also show experimentally that the classical conjugate gradient method, generally divergent for such indefinite problems, can converge when preconditioned with the overlapping Schwarz preconditioner.
Subject 65-XX

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