Convergence analysis of oversampled collocation boundary element methods in 2D

TL;DR

This paper investigates how oversampling in collocation boundary element methods can enhance convergence rates and robustness, sometimes surpassing Galerkin methods, especially in 2D Helmholtz problems.

Contribution

It demonstrates that oversampling can significantly improve convergence and robustness of collocation methods, with theoretical analysis and numerical validation for 2D Helmholtz equations.

Findings

Oversampling can lead to higher convergence rates than Galerkin methods.

Linear oversampling improves error at a cubic rate in the oversampling factor.

Oversampling reduces sensitivity to collocation point choices, ensuring convergence.

Abstract

Collocation boundary element methods for integral equations are easier to implement than Galerkin methods because the elements of the discretization matrix are given by lower-dimensional integrals. For that same reason, the matrix assembly also requires fewer computations. However, collocation methods typically yield slower convergence rates and less robustness, compared to Galerkin methods. We explore the extent to which oversampled collocation can improve both robustness and convergence rates. We show that in some cases convergence rates can actually be higher than the corresponding Galerkin method, although this requires oversampling at a faster than linear rate. In most cases of practical interest, oversampling at least lowers the error by a constant factor. This can still be a substantial improvement: we analyze an example where linear oversampling by a constant factor $J$ (leading to a rectangular system of size $JN \times N$) improves the error at a cubic rate in the constant $J$. Furthermore, the oversampled collocation method is much less affected by a poor choice of collocation points, as we show how oversampling can lead to guaranteed convergence. Numerical experiments are included for the two-dimensional Helmholtz equation.