Adaptive wavelet BEM for boundary integral equations: Theory and numerical experiments

TL;DR

This paper develops an adaptive wavelet boundary element method for solving boundary integral equations on manifolds, demonstrating theoretical convergence and practical efficiency through numerical experiments.

Contribution

It introduces a theoretically sound adaptive wavelet BEM approach for second kind Fredholm equations on manifolds, with verified convergence rates.

Findings

Adaptive methods improve efficiency in solving boundary integral equations.

Numerical results align with theoretical convergence predictions.

The approach effectively handles solutions with Besov regularity.

Abstract

In this paper, we are concerned with the numerical treatment of boundary integral equations by means of the adaptive wavelet boundary element method (BEM). In particular, we consider the second kind Fredholm integral equation for the double layer potential operator on patchwise smooth manifolds contained in $\mathbb{R}^3$. The corresponding operator equations are treated by means of adaptive implementations that are in complete accordance with the underlying theory. The numerical experiments demonstrate that adaptive methods really pay off in this setting. The observed convergence rates fit together very well with the theoretical predictions that can be made on the basis of a systematic investigation of the Besov regularity of the exact solution. Keywords: Besov spaces, weighted Sobolev spaces, adaptive wavelet BEM, non-linear approximation, integral equations, double layer potential operator, regularity, manifolds.