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 p\geq C_2 \log k, where C_1 is sufficiently small, and C_2 is sufficiently large  . 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^\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.
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 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 . 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^\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.
D
and
A
, and both are independent of k,h, and p