Helmholtz boundary integral methods and the pollution effect

TL;DR

This paper analyzes boundary integral methods for high-frequency Helmholtz problems, establishing how the degrees of freedom must grow with wavenumber and identifying which methods are affected by the pollution effect.

Contribution

It provides rigorous results on the growth of degrees of freedom needed for accuracy and identifies methods susceptible to the pollution effect in high-frequency Helmholtz boundary problems.

Findings

Determines the rate at which degrees of freedom must increase with wavenumber

Identifies which boundary integral methods suffer from the pollution effect

Provides first rigorous analysis for certain Galerkin and collocation methods

Abstract

This paper is concerned with solving the Helmholtz exterior Dirichlet and Neumann problems with large wavenumber $k$ and smooth obstacles using the standard second-kind boundary integral equations. We consider Galerkin and collocation methods -- with subspaces consisting of $\textit{either}$ piecewise polynomials (in 2-d for collocation, in any dimension for Galerkin) $\textit{or}$ trigonometric polynomials (in 2-d) -- as well as a fully discrete quadrature (Nystr\"om) method based on trigonometric polynomials (in 2-d). For each of these methods, we prove -- in many cases for the first time -- rigorous results about the fundamental question: how quickly must the number of degrees of freedom (the dimension of the approximation space) grow with $k$ to maintain accuracy of the computed solution? Importantly, we determine which of these methods suffer from $\textit{the pollution effect}$. That is, we address the question: must the number of points per wavelength $\to \infty$ to maintain accuracy as $k\to\infty$?