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Volume 28, Issue 1
Superconvergence of Gradient Recovery Schemes on Graded Meshes for Corner Singularities

Long Chen & Hengguang Li

DOI: 10.4208/jcm.2009.09-m1002

J. Comp. Math., 28 (2010), pp. 11-31.

Published online: 2010-02

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  • Abstract

For the linear finite element solution to the Poisson equation, we show that superconvergence exists for a type of graded meshes for corner singularities in polygonal domains. In particular, we prove that the $L^2$-projection from the piecewise constant field $∇u_N$ to the continuous and piecewise linear finite element space gives a better approximation of $∇u$ in the $H^1$-norm. In contrast to the existing superconvergence results, we do not assume high regularity of the exact solution.

  • Keywords

Superconvergence, Graded meshes, Weighted Sobolev spaces, Singular solutions, The finite element method, Gradient recovery schemes.

  • AMS Subject Headings

65N12, 65N30, 65N50.

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{JCM-28-11, author = {Long Chen and Hengguang Li}, title = {Superconvergence of Gradient Recovery Schemes on Graded Meshes for Corner Singularities}, journal = {Journal of Computational Mathematics}, year = {2010}, volume = {28}, number = {1}, pages = {11--31}, abstract = {

For the linear finite element solution to the Poisson equation, we show that superconvergence exists for a type of graded meshes for corner singularities in polygonal domains. In particular, we prove that the $L^2$-projection from the piecewise constant field $∇u_N$ to the continuous and piecewise linear finite element space gives a better approximation of $∇u$ in the $H^1$-norm. In contrast to the existing superconvergence results, we do not assume high regularity of the exact solution.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2009.09-m1002}, url = {http://global-sci.org/intro/article_detail/jcm/8504.html} }
TY - JOUR T1 - Superconvergence of Gradient Recovery Schemes on Graded Meshes for Corner Singularities AU - Long Chen & Hengguang Li JO - Journal of Computational Mathematics VL - 1 SP - 11 EP - 31 PY - 2010 DA - 2010/02 SN - 28 DO - http://doi.org/10.4208/jcm.2009.09-m1002 UR - https://global-sci.org/intro/article_detail/jcm/8504.html KW - Superconvergence, Graded meshes, Weighted Sobolev spaces, Singular solutions, The finite element method, Gradient recovery schemes. AB -

For the linear finite element solution to the Poisson equation, we show that superconvergence exists for a type of graded meshes for corner singularities in polygonal domains. In particular, we prove that the $L^2$-projection from the piecewise constant field $∇u_N$ to the continuous and piecewise linear finite element space gives a better approximation of $∇u$ in the $H^1$-norm. In contrast to the existing superconvergence results, we do not assume high regularity of the exact solution.

Long Chen and Hengguang Li. (2010). Superconvergence of Gradient Recovery Schemes on Graded Meshes for Corner Singularities. Journal of Computational Mathematics. 28 (1). 11-31. doi:10.4208/jcm.2009.09-m1002
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