A weakly overlapping domain decomposition preconditioner for the finite element solution of elliptic partial differential equations

We present anew two-level additive Schwarz domain decomposition preconditioner which is appropriate for use in the parallel finite element solution of elliptic partial differential equations ( PDEs). As with most parallel domain decomposition methods each processor may be assigned one or more subdomains, and the preconditioner is such that the processors are able to solve their own subproblem(s) concurrently. The novel feature of the technique proposed here is that it requires just a single layer of overlap in the elements which make up each subdomain at each level of refinement, and it is shown that this amount of overlap is sufficient to yield an optimal preconditioner. Some numerical experiments posed in both two and three space dimensions are included to confirm that the condition number when using the new preconditioner is indeed independent of the level of mesh refinement on the test problems considered.