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Volume 28, Issue 1
Uniform Superconvergence of a Finite Element Method with Edge Stabilization for Convection-Diffusion Problems

Sebastian Franz, Torsten Linß, Hans-Görg Roos & Sebastian Schiller

DOI: 10.4208/jcm.2009.09-m1005

J. Comp. Math., 28 (2010), pp. 32-44.

Published online: 2010-02

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  • Abstract

In the present paper the edge stabilization technique is applied to a convection-diffusion problem with exponential boundary layers on the unit square, using a Shishkin mesh with bilinear finite elements in the layer regions and linear elements on the coarse part of the mesh. An error bound is proved for ${||πu−u^h||}_E$ where $πu$ is some interpolant of the solution $u$ and $u^h$ the discrete solution. This supercloseness result implies an optimal error estimate with respect to the $L_2$ norm and opens the door to the application of postprocessing for improving the discrete solution.

  • Keywords

Convection-diffusion problems, Edge stabilization, FEM, Uniform convergence, Shishkin mesh.

  • AMS Subject Headings

65N15, 65N30, 65N50.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-28-32, author = {Franz , SebastianLinß , TorstenRoos , Hans-Görg and Schiller , Sebastian}, title = {Uniform Superconvergence of a Finite Element Method with Edge Stabilization for Convection-Diffusion Problems}, journal = {Journal of Computational Mathematics}, year = {2010}, volume = {28}, number = {1}, pages = {32--44}, abstract = {

In the present paper the edge stabilization technique is applied to a convection-diffusion problem with exponential boundary layers on the unit square, using a Shishkin mesh with bilinear finite elements in the layer regions and linear elements on the coarse part of the mesh. An error bound is proved for ${||πu−u^h||}_E$ where $πu$ is some interpolant of the solution $u$ and $u^h$ the discrete solution. This supercloseness result implies an optimal error estimate with respect to the $L_2$ norm and opens the door to the application of postprocessing for improving the discrete solution.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2009.09-m1005}, url = {http://global-sci.org/intro/article_detail/jcm/8505.html} }
TY - JOUR T1 - Uniform Superconvergence of a Finite Element Method with Edge Stabilization for Convection-Diffusion Problems AU - Franz , Sebastian AU - Linß , Torsten AU - Roos , Hans-Görg AU - Schiller , Sebastian JO - Journal of Computational Mathematics VL - 1 SP - 32 EP - 44 PY - 2010 DA - 2010/02 SN - 28 DO - http://doi.org/10.4208/jcm.2009.09-m1005 UR - https://global-sci.org/intro/article_detail/jcm/8505.html KW - Convection-diffusion problems, Edge stabilization, FEM, Uniform convergence, Shishkin mesh. AB -

In the present paper the edge stabilization technique is applied to a convection-diffusion problem with exponential boundary layers on the unit square, using a Shishkin mesh with bilinear finite elements in the layer regions and linear elements on the coarse part of the mesh. An error bound is proved for ${||πu−u^h||}_E$ where $πu$ is some interpolant of the solution $u$ and $u^h$ the discrete solution. This supercloseness result implies an optimal error estimate with respect to the $L_2$ norm and opens the door to the application of postprocessing for improving the discrete solution.

Franz , SebastianLinß , TorstenRoos , Hans-Görg and Schiller , Sebastian. (2010). Uniform Superconvergence of a Finite Element Method with Edge Stabilization for Convection-Diffusion Problems. Journal of Computational Mathematics. 28 (1). 32-44. doi:10.4208/jcm.2009.09-m1005
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