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 [Melenk-Sauter, 2010], [Melenk-Sauter, 2011], [Esterhazy-Melenk, 2012], [Melenk-Parsania-Sauter, 2013]. This theory shows that, if the solution operator is bounded polynomially in the wavenumber $k$, then the Galerkin method is quasioptimal provided that $hk/p leq C_1$ and $pgeq C_2 log k$, where $C_1$ is sufficiently small, $C_2$ is sufficiently large, and both are independent of $k,h,$ and $p$. This paper proves the analogous quasioptimality result for the heterogeneous (i.e. variable coefficient) Helmholtz equation, posed in $mathbb{R}^d$, $d=2,3$, with the Sommerfeld radiation condition at infinity, and $C^infty$ coefficients. We also prove a bound on the relative error of the Galerkin solution in the particular case of the plane-wave scattering problem.