Multilevel additive Schwarz method for the h-p version of the Galerkin boundary element method

We study a multilevel additive Schwarz method for the h-p version of the Galerkin boundary element method with geometrically graded meshes. Both hypersingular and weakly singular integral equations of the first kind are considered. As it is well known the h-p version with geometric meshes converges exponentially fast in the energy norm. However, the condition number of the Galerkin matrix in this case blows up exponentially in the number of unknowns M . We prove that the condition number κ(P ) of the multilevel additive Schwarz operator behaves like O( √ M logM). As a direct consequence of this we also give the results for the 2-level preconditioner and also for the h-p version with quasi-uniform meshes. Numerical results supporting our theory are presented.