Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods
type: Item
instance of: scholarly article
title: Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods
zbMATH Open document ID: 1058.65103
author: Mark Ainsworth
published in: Journal of Computational Physics
publication date: 2004-08-16T00:00:00Z
review text: Discontinuous finite element methods (DGFEM) are a promising approach for discretizing advection equations. The paper analyzes the dispersion and the dissipation error of \(N\)-th order DGFEM on partitions of \(\mathbb R^d\), \(d\in\{2,3\}\), consisting of tensor product mesh cells of size \(h^d\). In the case that the wave number \(k\) is small with respect to the mesh size, it is shown that the order of convergence of the DGFEM is higher than of the standard Galerkin FEM. Consideration of the leading order term of the relative errors leads to the suggestion that it is more efficient to increase \(N\) than to decrease \(h\). The case of fixed \(h\) and varying \(N\) is analyzed in detail. Choosing \(N\) such that \(2N+1 \approx \kappa hk\) with \(\kappa > 1\) is proposed to be the most efficient approach and an exponential order of convergence is proven in this case. The relative errors are related to certain Padé approximants of the exponential. The remainder in these Padé approximants are analyzed and they are linked to a certain eigenvalue problem.
reviewed by: Volker John
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 35L45
zbMATH DE Number: 2091353
zbMATH Keywords: discrete dispersion relation
zbMATH Keywords: high wave number
zbMATH Keywords: discontinuous Galerkin approximation
zbMATH Keywords: hp-finite element method
zbMATH Keywords: error bound
zbMATH Keywords: discontinuous finite element methods
zbMATH Keywords: advection equations
zbMATH Keywords: convergence
MaRDI profile type: Publication
full work available at URL: j.jcp.2004.01.004
OpenAlex ID: W2002083525
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DOI: 10.1016/J.JCP.2004.01.004
description: scientific article; zbMATH DE number 2091353