Local convergence of the boundary element method on polyhedral domains
Provides theoretical justification for local convergence behavior of boundary element methods on non-smooth domains, which is important for practitioners using these methods.
The paper proves local convergence rates for the boundary element method on polyhedral domains, showing that local rates are limited by local and global regularity. The analysis covers Symm's integral equation in L^2 and the hypersingular equation in H^1.
The local behavior of the lowest order boundary element method on quasi-uniform meshes for Symm's integral equation and the stabilized hyper-singular integral equation on polygonal/polyhedral Lipschitz domains is analyzed. We prove local a priori estimates in $L^2$ for Symm's integral equation and in $H^1$ for the hypersingular equation. The local rate of convergence is limited by the local regularity of the sought solution and the sum of the global regularity and additional regularity provided by the shift theorem for a dual problem.