Wavenumber-explicit convergence of the hp-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients
Résumé
A convergence theory for the $hp$-FEM applied to a variety of constant-coefficient Helmholtz problems was pioneered in the papers \cite{MeSa:10}, \cite{MeSa:11}, \cite{EsMe:12}, \cite{MePaSa:13}. 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 $p\geq C_2 \log k$, where $C_1$ is sufficiently small, $C_2$ is sufficiently large, and both are independent 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 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 $\Rea^d$, $d=2,3$, 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.
These are the first ever results on the wavenumber-explicit convergence of the $hp$-FEM for the Helmholtz equation with variable coefficients.