Domain Decomposition Algorithms for Indefinite Hypersingular Integral Equations: The h and p Versions

Abstract

Iterative methods for linear systems of algebraic equations arising from the h and p version boundary element discretizations of indefinite strongly elliptic integral equations are considered. The integral operators under consideration are strongly singular and therefore are pseudodifferential operators of order plus 1. Here we extend the approach introduced by Cai and Widlund [SIAM J. Sci. Statist. Comput., 13 (1992), pp. 243--258] for finite element discretizations to boundary element discretizations. We present an additive Schwarz method which is an efficient preconditioner for the GMRES, an iterative method of conjugate gradient type. The rate of convergence of the preconditioned GMRES algorithm is shown to be bounded for the h version and grow as log2 p for the p version.