Analysis of generalized Schwarz alternating procedure for domain decomposition

The Schwartz alternating method(SAM) is the theoretical basis for domain decomposition which itself is a powerful tool both for parallel computation and for computing in complicated domains. The convergence rate of the classical SAM is very sensitive to the overlapping size between each subdomain, which is not desirable for most applications. We propose a generalized SAM procedure which is an extension of the modified SAM proposed by P.-L. Lions. Instead of using only Dirichlet data at the artificial boundary between subdomains, we take a convex combination of u and {partial_derivative}u/{partial_derivative}n, i.e. {partial_derivative}u/{partial_derivative}n + {Lambda}u, where {Lambda} is some {open_quotes}positive{close_quotes} operator. Convergence of the modified SAM without overlapping in a quite general setting has been proven by P.-L.Lions using delicate energy estimates. The important questions remain for the generalized SAM. (1) What is the most essential mechanism for convergence without overlapping? (2) Given the partial differential equation, what is the best choice for the positive operator {Lambda}? (3) In the overlapping case, is the generalized SAM superior to the classical SAM? (4) What is the convergence rate and what does it depend on? (5) Numerically can we obtain an easy to implement operator {Lambda} such that the convergence is independent ofmore » the mesh size. To analyze the convergence of the generalized SAM we focus, for simplicity, on the Poisson equation for two typical geometry in two subdomain case.« less