2025/07/30 by Galkowski, Jeffrey, Rachh, Manas, Spence, Euan A.
#Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2507.22797
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 (BIEs) for these problems. We consider Galerkin and collocation methods - with subspaces consisting of either piecewise polynomials (in 2-d for collocation, in any dimension for Galerkin) or trigonometric polynomials (in 2-d) - as well as a fully discrete quadrature (a.k.a., Nyström) method based on trigonometric polynomials (in 2-d). For each of these methods, we address the fundamental question: how quickly must N, the dimension of the approximation space, grow with k to maintain accuracy as k→∞? For the methods involving piecewise-polynomials, we give sufficient conditions for k-uniform quasi-optimality. For the Galerkin method we show that these conditions are, in fact, necessary and sufficient. In particular we prove that the Galerkin method suffers from the pollution effect; i.e., N growing like kd-1 is often not sufficient for k-uniform quasi-optimality. For the Dirichlet BIEs, pollution occurs when the obstacle is trapping - and we also give numerical experiments illustrating this - but for the Neumann BIEs pollution occurs even when the obstacle is a ball. For all the methods involving trigonometric polynomials, we show that, up to potential factors of kε for any ε>0, there is no pollution (even for trapping obstacles). These are the first results about k-explicit convergence of collocation or Nyström methods applied to the Dirichlet BIEs, the first results about k-explicit convergence of any method used to solve the standard second-kind Neumann BIEs, and the first results proving that a boundary integral method applied to the Helmholtz equation suffers from the pollution effect.