Helmholtz boundary integral methods and the pollution effect
This addresses accuracy and efficiency challenges in computational wave scattering for applications like acoustics and electromagnetics, but it is incremental as it extends existing methods with new theoretical analysis.
The paper tackles the problem of solving Helmholtz exterior Dirichlet and Neumann problems with large wavenumbers using boundary integral methods, determining which methods suffer from the pollution effect and proving rigorous results on how the number of degrees of freedom must scale with wavenumber to maintain accuracy.
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 (Nyström) 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 $k\to\infty$?