Local convergence of the boundary element method on polyhedral domains

Abstract

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 for Symm’s integral equation and in for the hyper-singular equation. The local rate of convergence is limited by the local regularity of the sought solution and the sum of the rates given by the global regularity and additional regularity provided by the shift theorem for a dual problem.

Caccioppoli-type estimates and $$\mathcal {H}$$-matrix approximations to inverses for FEM-BEM couplings

“…A remedy for this problem is to localize the double-layer potential by splitting it into a local near-field and a non-local, but smooth far-field. This techniques follows [17], where a similar localization using commutators is employed.…”

Local convergence of the FEM for the integral fractional Laplacian

“…A key tool in the proof of a similar result for the BEM in [FM18] was the use of properties of the (singleor double-layer) potentials or, more precisely, a Caccioppoli type inequality. This interior regularity result allowed us to control derivatives of the potentials.…”

“…In this regard, fractional operators are similar to the integral operators appearing in the boundary element method (BEM), [SS11]. For the BEM, local error estimates and improved convergence results are available as well, see, e.g., [Sar87,Tra95,ST96,FM18], which differ from the ones for the FEM in the way that the error contribution in the weaker norm -sometimes called 'slush term' in the literature -is in a global norm instead of a local norm due to the non-local nature of the appearing operators.…”

Caccioppoli-type estimates and $\mathcal{H}$-Matrix approximations to inverses for FEM-BEM couplings

“…A remedy for this problem is to localize the double-layer potential by splitting it into a local near-field and a non-local, but smooth far-field. This techniques follows [FM18], where a similar localization using commutators is employed and a more detailed description of the method can be found.…”