An analysis of least-squares oversampled collocation methods for compactly perturbed boundary integral equations in two dimensions

TL;DR

This paper extends the analysis of least-squares oversampled collocation methods for boundary integral equations to cases with compact perturbations, demonstrating preserved convergence and advantages in 2D Laplace problems.

Contribution

It provides a comprehensive theoretical analysis of oversampled collocation methods under compact perturbations, confirming their effectiveness for boundary integral equations in 2D.

Findings

Optimal convergence rates are maintained under perturbations.

Sufficient oversampling rates are identified for stability.

Numerical experiments confirm theoretical predictions.

Abstract

In recent work (Maierhofer & Huybrechs, 2022, Adv. Comput. Math.), the authors showed that least-squares oversampling can improve the convergence properties of collocation methods for boundary integral equations involving operators of certain pseudo-differential form. The underlying principle is that the discrete method approximates a Bubnov$-$Galerkin method in a suitable sense. In the present work, we extend this analysis to the case when the integral operator is perturbed by a compact operator $\mathcal{K}$ which is continuous as a map on Sobolev spaces on the boundary, $\mathcal{K}:H^{p}\rightarrow H^{q}$ for all $p,q\in\mathbb{R}$. This study is complicated by the fact that both the test and trial functions in the discrete Bubnov-Galerkin orthogonality conditions are modified over the unperturbed setting. Our analysis guarantees that previous results concerning optimal convergence rates and sufficient rates of oversampling are preserved in the more general case. Indeed, for the first time, this analysis provides a complete explanation of the advantages of least-squares oversampled collocation for boundary integral formulations of the Laplace equation on arbitrary smooth Jordan curves in 2D. Our theoretical results are shown to be in very good agreement with numerical experiments.