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A convergence theory for the $hp$-FEM applied to a variety ofconstant-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 solutionoperator is bounded polynomially in the wavenumber $k$, then the Galerkinmethod is quasioptimal provided that $hk/p \leq C_1$ and $p\geq C_2 \log k$,where $C_1$ is sufficiently small, $C_2$ is sufficiently large, and both areindependent of $k,h,$ and $p$. The significance of this result is that if$hk/p= C_1$ and $p=C_2\log k$, then quasioptimality is achieved with the totalnumber of degrees of freedom proportional to $k^d$; i.e., the $hp$-FEM does notsuffer from the pollution effect. This paper proves the analogousquasioptimality result for the heterogeneous (i.e. variable-coefficient)Helmholtz equation, posed in $\mathbb{R}^d$, $d=2,3$, with the Sommerfeldradiation condition at infinity, and $C^\infty$ coefficients. We also prove abound on the relative error of the Galerkin solution in the particular case ofthe plane-wave scattering problem. These are the first ever results on thewavenumber-explicit convergence of the $hp$-FEM for the Helmholtz equation withvariable coefficients.

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