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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 = {}, 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 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 & Hengguang Li. (2019). 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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