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Volume 19, Issue 1
Recovery-Based a Posteriori Error Estimation for Elliptic Interface Problems Based on Partially Penalized Immersed Finite Element Methods

Yanping Chen, Zhirou Deng & Yunqing Huang

Int. J. Numer. Anal. Mod., 19 (2022), pp. 126-155.

Published online: 2022-03

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  • Abstract

This paper develops a recovery-based a posteriori error estimation for elliptic interface problems based on partially penalized immersed finite element (PPIFE) methods. Due to the low regularity of solution at the interface, standard gradient recovery methods cannot obtain superconvergent results. To overcome this drawback a new gradient recovery method is proposed that applies superconvergent cluster recovery (SCR) operator on each subdomain and weighted average (WA) operator at recovering points on the approximated interface. We prove that the recovered gradient superconverges to the exact gradient at the rate of $O(h^{1.5}).$ Consequently, the proposed method gives an asymptotically exact a posteriori error estimator for the PPIFE methods and the adaptive algorithm. Numerical examples show that the error estimator and the corresponding adaptive algorithm are both reliable and efficient.

  • Keywords

Interface problems, a posteriori error estimation, immersed finite element methods, gradient recovery, adaptive algorithm.

  • AMS Subject Headings

35R35, 65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-19-126, author = {Chen , YanpingDeng , Zhirou and Huang , Yunqing}, title = {Recovery-Based a Posteriori Error Estimation for Elliptic Interface Problems Based on Partially Penalized Immersed Finite Element Methods}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2022}, volume = {19}, number = {1}, pages = {126--155}, abstract = {

This paper develops a recovery-based a posteriori error estimation for elliptic interface problems based on partially penalized immersed finite element (PPIFE) methods. Due to the low regularity of solution at the interface, standard gradient recovery methods cannot obtain superconvergent results. To overcome this drawback a new gradient recovery method is proposed that applies superconvergent cluster recovery (SCR) operator on each subdomain and weighted average (WA) operator at recovering points on the approximated interface. We prove that the recovered gradient superconverges to the exact gradient at the rate of $O(h^{1.5}).$ Consequently, the proposed method gives an asymptotically exact a posteriori error estimator for the PPIFE methods and the adaptive algorithm. Numerical examples show that the error estimator and the corresponding adaptive algorithm are both reliable and efficient.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/20352.html} }
TY - JOUR T1 - Recovery-Based a Posteriori Error Estimation for Elliptic Interface Problems Based on Partially Penalized Immersed Finite Element Methods AU - Chen , Yanping AU - Deng , Zhirou AU - Huang , Yunqing JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 126 EP - 155 PY - 2022 DA - 2022/03 SN - 19 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/20352.html KW - Interface problems, a posteriori error estimation, immersed finite element methods, gradient recovery, adaptive algorithm. AB -

This paper develops a recovery-based a posteriori error estimation for elliptic interface problems based on partially penalized immersed finite element (PPIFE) methods. Due to the low regularity of solution at the interface, standard gradient recovery methods cannot obtain superconvergent results. To overcome this drawback a new gradient recovery method is proposed that applies superconvergent cluster recovery (SCR) operator on each subdomain and weighted average (WA) operator at recovering points on the approximated interface. We prove that the recovered gradient superconverges to the exact gradient at the rate of $O(h^{1.5}).$ Consequently, the proposed method gives an asymptotically exact a posteriori error estimator for the PPIFE methods and the adaptive algorithm. Numerical examples show that the error estimator and the corresponding adaptive algorithm are both reliable and efficient.

Yanping Chen, Zhirou Deng & Yunqing Huang. (2022). Recovery-Based a Posteriori Error Estimation for Elliptic Interface Problems Based on Partially Penalized Immersed Finite Element Methods. International Journal of Numerical Analysis and Modeling. 19 (1). 126-155. doi:
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