Multilevel Additive Schwarz Method for the Version of the Galerkin Boundary Element Method

We study a multilevel additive Schwarz method for the - 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 - 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 . We prove that the condition number of the multilevel additive Schwarz operator behaves like . As a direct consequence of this we also give the results for the -level preconditioner and also for the - version with quasi-uniform meshes. Numerical results supporting our theory are presented.