## 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 hk/p≤C_{1} and p≥C_{2}logk, 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}logk, 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.

Original language | English (US) |
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Pages (from-to) | 59-69 |

Number of pages | 11 |

Journal | Computers and Mathematics with Applications |

Volume | 113 |

DOIs | |

State | Published - May 1 2022 |

## Keywords

- Helmholtz equation
- High frequency
- Pollution effect
- hp-FEM

## ASJC Scopus subject areas

- Modeling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics