Geometry related convergence results for domain decomposition algorithms

For general second-order elliptic partial differential equations, the Schwarz alternating procedure is proved to converge at a rate independent of the aspect ratio for L-shaped, T-shaped, and C-shaped domains. The results cover both continuous and discrete versions of the Schwarz algorithm. Moreover, they apply to the nonoverlapping Schur complement algorithms with the preconditioner proposed in Chan. In particular, it is shown that the condition number of the preconditioned interface operator is bounded by 2 for all L-shaped and T-shaped domains. This improves similar geometry-independent convergence results for the Schur complement algorithms obtained previously by Chan and Resasco.