Generalized Schwarz alternating procedure for domain decomposition

Here we propose a generalized Swartz alternating method (SAM) procedure which is a generalization of the modified SAM proposed by P.-L. Lions in (15). Instead of using only Dirichlet data at the artificial boundaries between subdomains, we take a convex combination of u and the normal derivative ;For more general elliptic operators and more complicated geometries with two subdomains, we can use the equivalence of the elliptic operators and variable transformation to reduce them to the two simple cases. We can also extend this generalized SAM to the multi-domain case by reducing it to the two subdomain case. Motivated by the absorbing type of boundary conditions in (7), we use some local operators which may involve the tangential information to approximate the Dirichlet to Neumann operator in numerical calculation. These numerical schemes can be easily in-cooperated into the existing numerical scheme in domain decomposition to improve the performance since they do not change the data structure in the interior of the domain or subdomains. Finally some interpretation in some other domain decomposition context is presented.;In the appendix, an optimal domain decomposition cut using variational level set formulation is proposed. The cut will minimize the communication cost by cutting through the least number of edges of the finite elements and maintaining a balanced number of nodes (computation cost) in each subdomain. (Abstract shortened by UMI.).