Two-Level Additive Schwarz Preconditioners for the h-p Version of the Galerkin Boundary Element Method for 2-d Problems

Abstract

We study two-level additive Schwarz preconditioners for the h-p version of the Galerkin boundary element method when used to solve hypersingular integral equations of the first kind, which arise from the Neumann problems for the Laplacian in two dimensions. Overlapping and non-overlapping methods are considered. We prove that the non-overlapping preconditioner yields a system of equations having a condition number bounded by   where H _i is the length of the i-th subdomain, h _i is the maximum length of the elements in this subdomain, and p is the maximum polynomial degree used. For the overlapping method, we prove that the condition number is bounded by   where δ is the size of the overlap and H=max _i H _i . We also discuss the use of the non-overlapping method when the mesh is geometrically graded. The condition number in that case is bounded by clog² M, where M is the degrees of freedom.