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General DG-methods for highly indefinite Helmholtz problems


Melenk, Jens Markus; Parsani, Asieh; Sauter, Stefan A (2013). General DG-methods for highly indefinite Helmholtz problems. Journal of Scientific Computing, 57(3):536-581.

Abstract

We develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in Rd, d ∈ {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 √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.

Abstract

We develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in Rd, d ∈ {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 √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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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:December 2013
Deposited On:27 Dec 2013 13:02
Last Modified:05 Apr 2016 17:17
Publisher:Springer New York LLC
ISSN:0885-7474
Publisher DOI:https://doi.org/10.1007/s10915-013-9726-8

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