Volume 14, Issue 2
Superconvergent Cluster Recovery Method for the Crouzeix-Raviart Element

Yidan Zhang, Yaoyao Chen, Yunqing HuangNianyu Yi

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 508-526.

Published online: 2021-01

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  • Abstract

In this paper, we propose and numerically investigate a superconvergent cluster recovery (SCR) method for the Crouzeix-Raviart (CR) element. The proposed recovery method reconstructs a $C^0$ linear gradient. A linear polynomial approximation is obtained by a least square fitting to the CR element approximation at certain sample points, and then taken derivatives to obtain the recovered gradient. The SCR recovery operator is superconvergent on uniform mesh of four patterns. Numerical examples show that SCR can produce a superconvergent gradient approximation for the CR element, and provide an asymptotically exact error estimator in the adaptive CR finite element method.

  • Keywords

Crouzeix-Raviart element, gradient recovery, superconvergent cluster recovery.

  • AMS Subject Headings

65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-508, author = {Zhang , Yidan and Chen , Yaoyao and Huang , Yunqing and Yi , Nianyu}, title = {Superconvergent Cluster Recovery Method for the Crouzeix-Raviart Element}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2021}, volume = {14}, number = {2}, pages = {508--526}, abstract = {

In this paper, we propose and numerically investigate a superconvergent cluster recovery (SCR) method for the Crouzeix-Raviart (CR) element. The proposed recovery method reconstructs a $C^0$ linear gradient. A linear polynomial approximation is obtained by a least square fitting to the CR element approximation at certain sample points, and then taken derivatives to obtain the recovered gradient. The SCR recovery operator is superconvergent on uniform mesh of four patterns. Numerical examples show that SCR can produce a superconvergent gradient approximation for the CR element, and provide an asymptotically exact error estimator in the adaptive CR finite element method.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0117}, url = {http://global-sci.org/intro/article_detail/nmtma/18609.html} }
TY - JOUR T1 - Superconvergent Cluster Recovery Method for the Crouzeix-Raviart Element AU - Zhang , Yidan AU - Chen , Yaoyao AU - Huang , Yunqing AU - Yi , Nianyu JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 508 EP - 526 PY - 2021 DA - 2021/01 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2020-0117 UR - https://global-sci.org/intro/article_detail/nmtma/18609.html KW - Crouzeix-Raviart element, gradient recovery, superconvergent cluster recovery. AB -

In this paper, we propose and numerically investigate a superconvergent cluster recovery (SCR) method for the Crouzeix-Raviart (CR) element. The proposed recovery method reconstructs a $C^0$ linear gradient. A linear polynomial approximation is obtained by a least square fitting to the CR element approximation at certain sample points, and then taken derivatives to obtain the recovered gradient. The SCR recovery operator is superconvergent on uniform mesh of four patterns. Numerical examples show that SCR can produce a superconvergent gradient approximation for the CR element, and provide an asymptotically exact error estimator in the adaptive CR finite element method.

Yidan Zhang, Yaoyao Chen, Yunqing Huang & Nianyu Yi. (2021). Superconvergent Cluster Recovery Method for the Crouzeix-Raviart Element. Numerical Mathematics: Theory, Methods and Applications. 14 (2). 508-526. doi:10.4208/nmtma.OA-2020-0117
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