Jeffrey Galkowski

Martin Averseng, Jeffrey Galkowski, Euan A. Spence

The $h$-version of the finite-element method ($h$-FEM) applied to the high-frequency Helmholtz equation has been a classic topic in numerical analysis since the 1990s. It is now rigorously understood that (using piecewise polynomials of degree $p$ on a mesh of a maximal width $h$) the conditions "$(hk)^p ρ$ sufficiently small" and "$(hk)^{2p} ρ$ sufficiently small" guarantee, respectively, $k$-uniform quasioptimality (QO) and bounded relative error (BRE), where $ρ$ is the norm of the solution operator with $ρ\sim k$ for non-trapping problems. Empirically, these conditions are observed to be optimal in the context of $h$-FEM with a uniform mesh. This paper demonstrates that QO and BRE can be achieved using certain non-uniform meshes that violate the conditions above on $h$ and involve coarser meshes away from trapping and in the perfectly matched layer (PML). The main theorem details how varying the meshwidth in one region affects errors both in that region and elsewhere. One notable consequence is that, for any scattering problem (trapping or nontrapping), in the PML one only needs $hk$ to be sufficiently small; i.e. there is no pollution in the PML. The motivating idea for the analysis is that the Helmholtz data-to-solution map behaves differently depending on the locations of both the measurement and data, in particular, on the properties of billiards trajectories (i.e. rays) through these sets. Because of this, it is natural that the approximation requirements for finite-element spaces in a subset should depend on the properties of billiard rays through that set. Inserting this behaviour into the latest duality arguments for the FEM applied to the high-frequency Helmholtz equation allows us to retain detailed information about the influence of $\textit{both}$ the mesh structure $\textit{and}$ the behaviour of the true solution on local errors in FEM.

Jeffrey Galkowski, Eike H. Müller, Euan A. Spence

We consider solving the exterior Dirichlet problem for the Helmholtz equation with the $h$-version of the boundary element method (BEM) using the standard second-kind combined-field integral equations. We prove a new, sharp bound on how the number of GMRES iterations must grow with the wavenumber $k$ to have the error in the iterative solution bounded independently of $k$ as $k\rightarrow \infty$ when the boundary of the obstacle is analytic and has strictly positive curvature. To our knowledge, this result is the first-ever sharp bound on how the number of GMRES iterations depends on the wavenumber for an integral equation used to solve a scattering problem. We also prove new bounds on how $h$ must decrease with $k$ to maintain $k$-independent quasi-optimality of the Galerkin solutions as $k \rightarrow \infty$ when the obstacle is nontrapping.

Jeffrey Galkowski, Euan A. Spence

We consider the numerical solution of high-frequency scattering problems modeled by the Helmholtz equation with a bounded obstacle. Although the analysis of this problem dates back at least 50 years, over the past decade or so, tools and techniques from $\textit{semiclassical analysis}$ have provided a new perspective and been used to settle several long-standing open problems in this area. Semiclassical analysis works in phase space (i.e., position and frequency) and describes rigorously the extent to which solutions of high-frequency PDEs are dictated by the properties of the corresponding geometric-optic rays. The goals of the article are to (i) give a introduction to semiclassical analysis aimed at non-experts and (ii) showcase some of the numerical-analysis results about finite-element methods, boundary-element methods, and domain-decomposition methods obtained using semiclassical techniques.

Jeffrey Galkowski, Manas Rachh, Euan A. Spence

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 $k\to\infty$?