A Multilevel Additive Schwarz Method for the
h-p Version of the Galerkin Boundary Element Method
Norbert Heuer, Ernst P. Stephan, Thanh Tran
Math. Comp., 67 (222), 501-518, 1998.
Abstract
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 $\kappa(P)$ of the
multilevel additive Schwarz operator behaves like $O(\sqrt{M}\log^2M)$.
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.