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Journal of Scientific Computing

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01-12-2013

General DG-Methods for Highly Indefinite Helmholtz Problems

Authors: J. M. Melenk, A. Parsania, S. Sauter

Published in: Journal of Scientific Computing | Issue 3/2013

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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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The DG method can also be formulated for geometries with curved boundaries.

To see this, e.g., for $j=3$, we employ the interpolation inequality [36, (B.5)] to $\nabla ^{3}u$ to obtain

$$\begin{aligned} \left\| \nabla ^{3}u\right\| _{L^{\infty }( \widehat{K}) }\le C\left\| \nabla ^{3}u\right\| _{L^{2}( \widehat{K}) }^{1-d/\left( 2\left( s-3\right) \right) }\left\| \nabla ^{3} u\right\| _{H^{s-3}( \widehat{K}) }^{d/\left( 2\left( s-3\right) \right) }\quad \forall u\in H^{s}( \widehat{K}) \end{aligned}$$

since $s>3+d/2$. The combination with (7.6) yields the desired bound in (7.7).

For a face $f$, the face normal $n_{f}:\partial f\rightarrow \mathbb S _{2}$ is defined to have length $1$, lies in the plane of $f$, and points to the exterior of $f$. The face normal derivative on $\partial f$ is then given by $\partial _{n_{f}}:=\left\langle n_{f},\nabla \cdot \right\rangle $.

The condition $p \ge j$ can be dropped if $E_{1,e} u$ vanishes to higher order at the vertex (0,1) due to appropriate assumptions on the function $w$.

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Title: General DG-Methods for Highly Indefinite Helmholtz Problems
Authors: J. M. Melenk
A. Parsania
S. Sauter
Publication date: 01-12-2013
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 3/2013
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-013-9726-8

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