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Article overview
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Wavenumber-explicit convergence of the $hp$-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients | | David Lafontaine
; Euan A. Spence
; Jared Wunsch
; | | Date: |
1 Oct 2020 | | Abstract: | A convergence theory for the $hp$-FEM applied to a variety of
constant-coefficient Helmholtz problems was pioneered in the papers
[Melenk-Sauter, 2010], [Melenk-Sauter, 2011], [Esterhazy-Melenk, 2012],
[Melenk-Parsania-Sauter, 2013]. This theory shows that, if the solution
operator is bounded polynomially in the wavenumber $k$, then the Galerkin
method is quasioptimal provided that $hk/p leq C_1$ and $pgeq C_2 log k$,
where $C_1$ is sufficiently small, and $C_2$ is sufficiently large.
This paper proves the analogous quasioptimality result for the heterogeneous
(i.e. variable coefficient) Helmholtz equation, posed in $mathbb{R}^d$,
$d=2,3$, with the Sommerfeld radiation condition at infinity, and $C^infty$
coefficients. We also prove a bound on the relative error of the Galerkin
solution in the particular case of the plane-wave scattering problem. | | Source: | arXiv, 2010.00585 | | Services: | Forum | Review | PDF | Favorites | |
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