TY - JOUR

T1 - Wavenumber-explicit convergence of the hp-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients

AU - Lafontaine, D.

AU - Spence, E. A.

AU - Wunsch, J.

N1 - Funding Information:
The authors thank Martin Averseng (ETH Zürich) and an anonymous referee for highlighting simplifications of the arguments in a earlier version of the paper. We also thank Théophile Chaumont-Frelet (INRIA, Nice) for useful discussions about the results of [35] , [36] . DL and EAS acknowledge support from EPSRC grant EP/1025995/1 . JW was partly supported by Simons Foundation grant 631302 .
Publisher Copyright:
© 2022 Elsevier Ltd

PY - 2022/5/1

Y1 - 2022/5/1

N2 - 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 hk/p≤C1 and p≥C2logk, where C1 is sufficiently small, C2 is sufficiently large, and both are independent of k,h, and p. The significance of this result is that if hk/p=C1 and p=C2logk, then quasioptimality is achieved with the total number of degrees of freedom proportional to kd; 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 Rd, 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.

AB - 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 hk/p≤C1 and p≥C2logk, where C1 is sufficiently small, C2 is sufficiently large, and both are independent of k,h, and p. The significance of this result is that if hk/p=C1 and p=C2logk, then quasioptimality is achieved with the total number of degrees of freedom proportional to kd; 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 Rd, 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.

KW - Helmholtz equation

KW - High frequency

KW - Pollution effect

KW - hp-FEM

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U2 - 10.1016/j.camwa.2022.03.007

DO - 10.1016/j.camwa.2022.03.007

M3 - Article

AN - SCOPUS:85126605484

SN - 0898-1221

VL - 113

SP - 59

EP - 69

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

ER -