Volume 26, Issue 4
A Posteriori Energy-Norm Error Estimates for Advection-Diffusion Equations Approximated by Weighted Interior Penalty Methods

Alexandre Ern & Annette F. Stephansen

DOI:

J. Comp. Math., 26 (2008), pp. 488-510

Published online: 2008-08

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  • Abstract

We propose and analyze a posteriori energy-norm error estimates for weighted interior penalty discontinuous Galerkin approximations of advection-diffusion-reaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diffusion tensor and are used to formulate the consistency terms in the discontinuous Galerkin method. The error upper bounds, in which all the constants are specified, consist of three terms: a residual estimator which depends only on the elementwise fluctuation of the discrete solution residual, a diffusive flux estimator where the weights used in the method enter explicitly, and a non-conforming estimator which is nonzero because of the use of discontinuous finite element spaces. The three estimators can be bounded locally by the approximation error. A particular attention is given to the dependency on problem parameters of the constants in the local lower error bounds. For moderate advection, it is shown that full robustness with respect to diffusion heterogeneities is achieved owing to the specific design of the weights in the discontinuous Galerkin method, while diffusion anisotropies remain purely local and impact the constants through the square root of the condition number of the diffusion tensor. For dominant advection, it is shown, in the spirit of previous work by Verfurth on continuous finite elements, that the local lower error bounds can be written with constants involving a cut-off for the ratio of local mesh size to the reciprocal of the square root of the lowest local eignevalue of the diffusion tensor.

  • Keywords

Discontinuous Galerkin Weighted interior penalty A posteriori error estimate Heterogeneous diffusion Advection-diffusion

  • AMS Subject Headings

65N30 65N15 76R99.

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@Article{JCM-26-488, author = {}, title = {A Posteriori Energy-Norm Error Estimates for Advection-Diffusion Equations Approximated by Weighted Interior Penalty Methods}, journal = {Journal of Computational Mathematics}, year = {2008}, volume = {26}, number = {4}, pages = {488--510}, abstract = { We propose and analyze a posteriori energy-norm error estimates for weighted interior penalty discontinuous Galerkin approximations of advection-diffusion-reaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diffusion tensor and are used to formulate the consistency terms in the discontinuous Galerkin method. The error upper bounds, in which all the constants are specified, consist of three terms: a residual estimator which depends only on the elementwise fluctuation of the discrete solution residual, a diffusive flux estimator where the weights used in the method enter explicitly, and a non-conforming estimator which is nonzero because of the use of discontinuous finite element spaces. The three estimators can be bounded locally by the approximation error. A particular attention is given to the dependency on problem parameters of the constants in the local lower error bounds. For moderate advection, it is shown that full robustness with respect to diffusion heterogeneities is achieved owing to the specific design of the weights in the discontinuous Galerkin method, while diffusion anisotropies remain purely local and impact the constants through the square root of the condition number of the diffusion tensor. For dominant advection, it is shown, in the spirit of previous work by Verfurth on continuous finite elements, that the local lower error bounds can be written with constants involving a cut-off for the ratio of local mesh size to the reciprocal of the square root of the lowest local eignevalue of the diffusion tensor.}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8639.html} }
TY - JOUR T1 - A Posteriori Energy-Norm Error Estimates for Advection-Diffusion Equations Approximated by Weighted Interior Penalty Methods JO - Journal of Computational Mathematics VL - 4 SP - 488 EP - 510 PY - 2008 DA - 2008/08 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8639.html KW - Discontinuous Galerkin KW - Weighted interior penalty KW - A posteriori error estimate KW - Heterogeneous diffusion KW - Advection-diffusion AB - We propose and analyze a posteriori energy-norm error estimates for weighted interior penalty discontinuous Galerkin approximations of advection-diffusion-reaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diffusion tensor and are used to formulate the consistency terms in the discontinuous Galerkin method. The error upper bounds, in which all the constants are specified, consist of three terms: a residual estimator which depends only on the elementwise fluctuation of the discrete solution residual, a diffusive flux estimator where the weights used in the method enter explicitly, and a non-conforming estimator which is nonzero because of the use of discontinuous finite element spaces. The three estimators can be bounded locally by the approximation error. A particular attention is given to the dependency on problem parameters of the constants in the local lower error bounds. For moderate advection, it is shown that full robustness with respect to diffusion heterogeneities is achieved owing to the specific design of the weights in the discontinuous Galerkin method, while diffusion anisotropies remain purely local and impact the constants through the square root of the condition number of the diffusion tensor. For dominant advection, it is shown, in the spirit of previous work by Verfurth on continuous finite elements, that the local lower error bounds can be written with constants involving a cut-off for the ratio of local mesh size to the reciprocal of the square root of the lowest local eignevalue of the diffusion tensor.
Alexandre Ern & Annette F. Stephansen. (1970). A Posteriori Energy-Norm Error Estimates for Advection-Diffusion Equations Approximated by Weighted Interior Penalty Methods. Journal of Computational Mathematics. 26 (4). 488-510. doi:
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