Wavenumber-explicit convergence of the hp-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients. (1st May 2022) - Lafontaine, D.; Spence, E.A.; Wunsch, J.
Abstract: 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 h k / p ≤ C 1 and p ≥ 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 h k / 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 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.