Sharp Preasymptotic Error Bounds for the Helmholtz [math]-FEM

Abstract

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 1-22, February 2025.
Abstract. In the analysis of the [math]-version of the finite-element method (FEM), with fixed polynomial degree [math], applied to the Helmholtz equation with wavenumber [math], the asymptotic regime is when [math] is sufficiently small and the sequence of Galerkin solutions are quasioptimal; here [math] is the [math] norm of the Helmholtz solution operator, with [math] for nontrapping problems. In the preasymptotic regime, one expects that if [math] is sufficiently small, then (for physical data) the relative error of the Galerkin solution is controllably small. In this paper, we prove the natural error bounds in the preasymptotic regime for the variable-coefficient Helmholtz equation in the exterior of a Dirichlet, or Neumann, or penetrable obstacle (or combinations of these) and with the radiation condition either realized exactly using the Dirichlet-to-Neumann map on the boundary of a ball or approximated either by a radial perfectly matched layer (PML) or an impedance boundary condition. Previously, such bounds for [math] were only available for Dirichlet obstacles with the radiation condition approximated by an impedance boundary condition. Our result is obtained via a novel generalization of the “elliptic-projection” argument (the argument used to obtain the result for [math]), which can be applied to a wide variety of abstract Helmholtz-type problems.