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Volume 38, Issue 1
Superconvergence Analysis of the Polynomial Preserving Recovery for Elliptic Problems with Robin Boundary Conditions

Yu Du, Haijun Wu & Zhimin Zhang

DOI: 10.4208/jcm.1911-m2018-0176

J. Comp. Math., 38 (2020), pp. 223-238.

Published online: 2020-02

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  • Abstract

We analyze the superconvergence property of the linear finite element method based on the polynomial preserving recovery (PPR) for Robin boundary elliptic problems on triangulations. First, we improve the convergence rate between the finite element solution and the linear interpolation under the $H^1$-norm by introducing a class of meshes satisfying the $Condition$ $(\alpha,\sigma,\mu)$. Then we prove the superconvergence of the recovered gradients post-processed by PPR and define an asymptotically exact a posteriori error estimator. Finally, numerical tests are provided to verify the theoretical findings.

  • Keywords

Superconvergence, Polynomial preserving recovery, Finite element methods, Robin boundary condition.

  • AMS Subject Headings

65N12, 65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

duyu87@csrc.ac.cn (Yu Du)

hjw@nju.edu.cn (Haijun Wu)

zmzhang@csrc.ac.cn (Zhimin Zhang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-38-223, author = {Du , YuWu , Haijun and Zhang , Zhimin}, title = {Superconvergence Analysis of the Polynomial Preserving Recovery for Elliptic Problems with Robin Boundary Conditions}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {38}, number = {1}, pages = {223--238}, abstract = {

We analyze the superconvergence property of the linear finite element method based on the polynomial preserving recovery (PPR) for Robin boundary elliptic problems on triangulations. First, we improve the convergence rate between the finite element solution and the linear interpolation under the $H^1$-norm by introducing a class of meshes satisfying the $Condition$ $(\alpha,\sigma,\mu)$. Then we prove the superconvergence of the recovered gradients post-processed by PPR and define an asymptotically exact a posteriori error estimator. Finally, numerical tests are provided to verify the theoretical findings.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1911-m2018-0176}, url = {http://global-sci.org/intro/article_detail/jcm/13692.html} }
TY - JOUR T1 - Superconvergence Analysis of the Polynomial Preserving Recovery for Elliptic Problems with Robin Boundary Conditions AU - Du , Yu AU - Wu , Haijun AU - Zhang , Zhimin JO - Journal of Computational Mathematics VL - 1 SP - 223 EP - 238 PY - 2020 DA - 2020/02 SN - 38 DO - http://doi.org/10.4208/jcm.1911-m2018-0176 UR - https://global-sci.org/intro/article_detail/jcm/13692.html KW - Superconvergence, Polynomial preserving recovery, Finite element methods, Robin boundary condition. AB -

We analyze the superconvergence property of the linear finite element method based on the polynomial preserving recovery (PPR) for Robin boundary elliptic problems on triangulations. First, we improve the convergence rate between the finite element solution and the linear interpolation under the $H^1$-norm by introducing a class of meshes satisfying the $Condition$ $(\alpha,\sigma,\mu)$. Then we prove the superconvergence of the recovered gradients post-processed by PPR and define an asymptotically exact a posteriori error estimator. Finally, numerical tests are provided to verify the theoretical findings.

Du , YuWu , Haijun and Zhang , Zhimin. (2020). Superconvergence Analysis of the Polynomial Preserving Recovery for Elliptic Problems with Robin Boundary Conditions. Journal of Computational Mathematics. 38 (1). 223-238. doi:10.4208/jcm.1911-m2018-0176
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