Domain decomposition algorithms for indefinite weakly singular integral equations: The h and p versions

We extend the approach of Cai and Widlund (Domain decomposition algorithms for indefinite elliptic problems, SIAM J. Sci. Stat. Comput. 13 (1992), 243-258), which was designed for finite element discretizations, to boundary element discretizations of indefinite weakly singular integral equations. Both the h and p versions of the Galerkin approximation are considered. We prove that the additive Schwarz method suggested by Cai and Widlund can be used for this equation as an efficient preconditioner for GMRES, an iterative method of conjugate gradient type. For both versions, the rates of convergence of this iterative method are shown to approach 1 only logarithmically as the degrees of freedom tend to infinity.