An Optimal Two-Level Overlapping Domain Decomposition Method for Elliptic Problems in Two and Three Dimensions

Abstract

The solution of linear systems of algebraic equations that arise from elliptic finite element problems is considered. A two-level overlapping domain decomposition method that can be viewed as a combination of the additive and multiplicative Schwarz methods is studied. This method combines the advantages of the two methods. It converges faster than the additive Schwarz algorithm and is more parallelizable than the multiplicative Schwarz algorithm, and works for general, not necessarily self-adjoint, linear, second-order, elliptic equations. The GMRES method is used to solve the resulting preconditioned linear system of equations and it is shown that the algorithm is optimal in the sense that the rate of convergence is independent of the mesh size and the number of subregions in both $R^2 $ and $R^3 $. A numerical comparison with the additive and multiplicative Schwarz preconditioned GMRES is reported.