Robust Adaptive $hp$ Discontinuous Galerkin Finite Element Methods for the Helmholtz Equation
Abstract
This paper presents an $hp$ a posteriori error analysis for the 2D Helmholtz equation that is robust in the polynomial degree $p$ and the wave number $k$. For the discretization, we consider a discontinuous Galerkin formulation that is unconditionally well posed. The a posteriori error analysis is based on the technique of equilibrated fluxes applied to a shifted Poisson problem, with the error due to the nonconformity of the discretization controlled by a potential reconstruction. We prove that the error estimator is both reliable and efficient, under the condition that the initial mesh size and polynomial degree are chosen such that the discontinuous Galerkin formulation converges, i.e., it is out of the regime of pollution. We confirm the efficiency of an $hp$-adaptive refinement strategy based on the presented robust a posteriori error estimator via several numerical examples.
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Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: A1121 - A1147
DOI: 10.1137/18M1207909
ISSN (online): 1095-7197
Copyright
© 2019, Society for Industrial and Applied Mathematics.
History
Submitted: 16 August 2018
Accepted: 29 January 2019
Published online: 11 April 2019
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Funding Information
Austrian Science Fund https://doi.org/10.13039/501100002428 : F 65, P 29197-N32
Vienna Science and Technology Fund https://doi.org/10.13039/501100001821 : MA14-006
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