Wavenumber-explicit convergence of the hp-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients
D. Lafontaine, Euan A. Spence, Jared Wunsch
Abstract
A convergence theory for the hp-FEM applied to a variety of constant-coefficient Helmholtz problems was pioneered in the papers [35], [36], [15], [34]. This theory shows that, if the solution operator is bounded polynomially in the wavenumber k, then the Galerkin method is quasioptimal provided that hk/p≤C1 and p≥C2logk, where C1 is sufficiently small, C2 is sufficiently large, and both are independent of k,h, and p. The significance of this result is that if hk/p=C1 and p=C2logk, then quasioptimality is achieved with the total number of degrees of freedom proportional to kd; i.e., the hp-FEM does not suffer from the pollution effect. This paper proves the analogous quasioptimality result for the heterogeneous (i.e. variable-coefficient) Helmholtz equation, posed in Rd, d=2,3, with the Sommerfeld radiation condition at infinity, and C∞ coefficients. We also prove a bound on the relative error of the Galerkin solution in the particular case of the plane-wave scattering problem. These are the first ever results on the wavenumber-explicit convergence of the hp-FEM for the Helmholtz equation with variable coefficients.