# Local convergence of the boundary element method on polyhedral domains

## TL;DR

This paper analyzes the local convergence properties of the lowest order boundary element method on polyhedral domains, providing local a priori estimates and convergence rates based on solution regularity.

## Contribution

It offers new local a priori estimates and convergence rate analysis for boundary element methods on polyhedral domains, considering solution regularity.

## Key findings

- Established local $L^2$ estimates for Symm's integral equation.
- Derived local $H^1$ estimates for hypersingular integral equations.
- Identified the influence of solution regularity on convergence rates.

## 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 $L^2$ 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.