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Abstract
A convergence theory for the hpFEM applied to a variety of constantcoefficient 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 hpFEM does not suffer from the pollution effect. This paper proves the analogous quasioptimality result for the heterogeneous (i.e. variablecoefficient) 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 planewave scattering problem. These are the first ever results on the wavenumberexplicit convergence of the hpFEM for the Helmholtz equation with variable coefficients.
Original language  English 

Pages (fromto)  5969 
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 ChaumontFrelet (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
 hpFEM
ASJC Scopus subject areas
 Modelling and Simulation
 Computational Theory and Mathematics
 Computational Mathematics
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 1 Finished

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