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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 2logk, 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 2logk, 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 |
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Pages (from-to) | 59-69 |
Number of pages | 11 |
Journal | Computers & Mathematics with Applications |
Volume | 113 |
Early online date | 18 Mar 2022 |
DOIs | |
Publication status | Published - 1 May 2022 |
Bibliographical note
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 .
Keywords
- Helmholtz equation
- High frequency
- Pollution effect
- hp-FEM
ASJC Scopus subject areas
- Modelling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics
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At the interface between semiclassical analysis and numerical analysis of Wave propogation problems
Spence, E. (PI)
Engineering and Physical Sciences Research Council
1/10/17 → 30/09/23
Project: Research council