Does the Helmholtz Boundary Element Method Suffer from the Pollution Effect?

SIAM Review, Volume 65, Issue 3, Page 806-828, August 2023.
In $d$ dimensions, accurately approximating an arbitrary function oscillating with frequency $\lesssim k$ requires $\sim$$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 $k→∞$, 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) (where accuracy is increased by decreasing the meshwidth $h$ and keeping the polynomial degree $p$ fixed) suffers from the pollution effect, and research over the last $\sim$30 years has resulted in a near-complete rigorous understanding of how quickly the number of degrees of freedom must grow with $k$ to maintain accuracy (and how this depends on both $p$ and properties of the scatterer). In contrast to the $h$-FEM, at least empirically, the $h$-version of the boundary element method (BEM) does not suffer from the pollution effect (recall that in the boundary element method the scattering problem is reformulated as an integral equation on the boundary of the scatterer, with this integral equation then solved numerically using a finite element--type approximation space). However, the current best results in the literature on how quickly the number of degrees of freedom for the $h$-BEM must grow with $k$ to maintain accuracy fall short of proving this. In this paper, we prove that the $h$-version of the Galerkin method applied to the standard second-kind boundary integral equations for solving the Helmholtz exterior Dirichlet problem does not suffer from the pollution effect when the obstacle is nontrapping (i.e., does not trap geometric-optic rays). While the proof of this result relies on information about the large-$k$ behavior of Helmholtz solution operators, we show in an appendix how the result can be proved using only Fourier series and asymptotics of Hankel and Bessel functions when the obstacle is a 2-d ball.