Wavenumber-explicit convergence of the hp-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients

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≥C₂log⁡k, 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=C₂log⁡k, then quasioptimality is achieved with the total number of degrees of freedom proportional to k^d; 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 R^d, 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.