The Helmholtz boundary element method does not suffer from the pollution effect

 Euan Spence (University of Bath)

Abstract. In $d$ dimensions, approximating an arbitrary function oscillating with frequency less than or equal to $k$ requires ~$k^d$ degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber $k$ and in $d$ dimensions) suffers from the pollution effect if, as $k$ increases, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold (i.e., faster than $k^d$ for domain-based formulations, such as finite element methods, and $k^{d-1}$ for boundary-based formulations, such as boundary element methods).

It is well known that the h-version of the finite element method (FEM) suffers from the pollution effect. In contrast, at least empirically, the h-version of the boundary element method (BEM) does not suffer from the pollution effect, but this has not been proved up till now.

In this talk, I will discuss recent results (obtained with Jeffrey Galkowski) showing that the h-BEM does not suffer from the pollution effect in certain common situations.