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Wavenumber-explicit convergence of the hp-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients.
- Source :
-
Computers & Mathematics with Applications . May2022, Vol. 113, p59-69. 11p. - Publication Year :
- 2022
-
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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *DEGREES of freedom
*GALERKIN methods
*HELMHOLTZ equation
*WAVENUMBER
Subjects
Details
- Language :
- English
- ISSN :
- 08981221
- Volume :
- 113
- Database :
- Academic Search Index
- Journal :
- Computers & Mathematics with Applications
- Publication Type :
- Academic Journal
- Accession number :
- 156129294
- Full Text :
- https://doi.org/10.1016/j.camwa.2022.03.007