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. These 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 [SIAM J. Numer. Anal., 24 (1987), pp. 382–390]. 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 [“Advances in Computer Methods for Partial Differential Equations, VI,” R. Vichnevetsky and R. S. Stepleman, eds., pp. 317–322].