Volume 8, Issue 4
Unified a Priori Error Estimate and a Posteriori Error Estimate of CIP-FEM for Elliptic Equations

Jianye Wang & Rui Ma

Adv. Appl. Math. Mech., 8 (2016), pp. 517-535.

Published online: 2018-05

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  • Abstract

This paper is devoted to a unified a priori and a posteriori error analysis of CIP-FEM (continuous interior penalty finite element method) for second-order elliptic problems. Compared with the classic a priori error analysis in literature, our technique can easily apply for any type regularity assumption on the exact solution, especially for the case of lower H1+s weak regularity under consideration, where 0 ≤ s ≤ 1/2. Because of the penalty term used in the CIP-FEM, Galerkin orthogonality is lost and C´ea Lemma for conforming finite element methods can not be applied immediately when 0≤s≤1/2. To overcome this difficulty, our main idea is introducing an auxiliary C 1 finite element space in the analysis of the penalty term. The same tool is also utilized in the explicit a posteriori error analysis of CIP-FEM.

  • Keywords

Finite element methods, continuous interior penalty, priori error estimate, posteriori error analysis.

  • AMS Subject Headings

65N30, 65N15, 35J25

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-8-517, author = {}, title = {Unified a Priori Error Estimate and a Posteriori Error Estimate of CIP-FEM for Elliptic Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {8}, number = {4}, pages = {517--535}, abstract = {

This paper is devoted to a unified a priori and a posteriori error analysis of CIP-FEM (continuous interior penalty finite element method) for second-order elliptic problems. Compared with the classic a priori error analysis in literature, our technique can easily apply for any type regularity assumption on the exact solution, especially for the case of lower H1+s weak regularity under consideration, where 0 ≤ s ≤ 1/2. Because of the penalty term used in the CIP-FEM, Galerkin orthogonality is lost and C´ea Lemma for conforming finite element methods can not be applied immediately when 0≤s≤1/2. To overcome this difficulty, our main idea is introducing an auxiliary C 1 finite element space in the analysis of the penalty term. The same tool is also utilized in the explicit a posteriori error analysis of CIP-FEM.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2014.m834}, url = {http://global-sci.org/intro/article_detail/aamm/12101.html} }
TY - JOUR T1 - Unified a Priori Error Estimate and a Posteriori Error Estimate of CIP-FEM for Elliptic Equations JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 517 EP - 535 PY - 2018 DA - 2018/05 SN - 8 DO - http://dor.org/10.4208/aamm.2014.m834 UR - https://global-sci.org/intro/aamm/12101.html KW - Finite element methods, continuous interior penalty, priori error estimate, posteriori error analysis. AB -

This paper is devoted to a unified a priori and a posteriori error analysis of CIP-FEM (continuous interior penalty finite element method) for second-order elliptic problems. Compared with the classic a priori error analysis in literature, our technique can easily apply for any type regularity assumption on the exact solution, especially for the case of lower H1+s weak regularity under consideration, where 0 ≤ s ≤ 1/2. Because of the penalty term used in the CIP-FEM, Galerkin orthogonality is lost and C´ea Lemma for conforming finite element methods can not be applied immediately when 0≤s≤1/2. To overcome this difficulty, our main idea is introducing an auxiliary C 1 finite element space in the analysis of the penalty term. The same tool is also utilized in the explicit a posteriori error analysis of CIP-FEM.

Jianye Wang & Rui Ma. (2020). Unified a Priori Error Estimate and a Posteriori Error Estimate of CIP-FEM for Elliptic Equations. Advances in Applied Mathematics and Mechanics. 8 (4). 517-535. doi:10.4208/aamm.2014.m834
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