Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems,part I: Algorithms and numerical results

Summary We describe sequential and parallel algorithms based on the Schwarz alternating method for the solution of mixed finite element discretizations of elliptic problems using the Raviart-Thomas finite element spaces. These lead to symmetric indefinite linear systems and the algorithms have some similarities with the traditional block Gauss-Seidel or block Jacobi methods with overlapping blocks. The indefiniteness requires special treatment. The sub-blocks used in the algorithm correspond to problems on a coarse grid and some overlapping subdomains and is based on a similar partition used in an algorithm of Dryja and Widlund for standard elliptic problems. If there is sufficient overlap between the subdomains, the algorithm converges with a rate independent of the mesh size, the number of subdomains and discontinuities of the coefficients. Extensions of the above algorithms to the case of local grid refinement is also described. Convergence theory for these algorithms will be presented in a subsequent paper.This work was supported in part by the National Science Foundation under Grant NSF-CCR-8903003, while the author was a graduate student at New York University, and in part by the Army Research Office under Grant DAAL 03-91-G-0150, while the author was a Visiting Assistant Researcher at UCLA