Local error estimates and post processing for the Galerkin boundary element method on polygons

TL;DR

This work addresses local error behavior of the Galerkin boundary element method for strongly elliptic pseudodifferential equations on polygonal boundaries. It develops local regularity results and explicit local Sobolev error bounds that separate corner-induced limitations from smoother subregions, highlighting how global negative-norm estimates influence local convergence. It then introduces a K-operator post-processing, defined as a spline-based convolution $K_h(v)$, which can substantially improve local accuracy (up to $O(h^{3})$) on subsegments when paired with a uniform mesh on a neighborhood and possibly graded meshes near corners. Numerical experiments on an L-shaped polygon corroborate the theory and demonstrate practical gains from K-operator post-processing in local convergence.

Abstract

In this paper we give local error estimates in Sobolev norms for the Galerkin method applied to strongly elliptic pseudodifferential equations on a polygon. By using the K-operator, an operator which averages the values of the Galerkin solution, we construct improved approximations.

Theorems & Definitions (20)

• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8
• Lemma 2.9
• Lemma 2.10