General DG-Methods for Highly Indefinite Helmholtz Problems |
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Authors: | J M Melenk A Parsania S Sauter |
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Affiliation: | 1. Institute for Analysis and Scientific Computing, Technische Universit?t Wien, Wiedner Hauptstrasse 8-10, 1040, Vienna, Austria 2. Institut für Mathematik, Universit?t Zürich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
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Abstract: | We develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in $\mathbb R ^{d}$ , $d\in \{1,2,3\}$ . The theory covers conforming as well as non-conforming generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, it is proved that the DG-method admits a unique solution under much weaker conditions. As an application we present the error analysis for the $hp$ -version of the finite element method explicitly in terms of the mesh width $h$ , polynomial degree $p$ and wavenumber $k$ . It is shown that the optimal convergence order estimate is obtained under the conditions that $kh/\sqrt{p}$ is sufficiently small and the polynomial degree $p$ is at least $O(\log k)$ . On regular meshes, the first condition is improved to the requirement that $kh/p$ be sufficiently small. |
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