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Unified a Priori Error Estimate and a Posteriori Error Estimate of CIP-FEM for Elliptic Equations

Part of: Elliptic equations and systems Partial differential equations, boundary value problems

Published online by Cambridge University Press:  27 May 2016

Jianye Wang  and
Rui Ma
Jianye Wang*
Affiliation:
LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China
Rui Ma*
Affiliation:
LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China
*
*Corresponding author. Email:jianguoyeluo@163.com (J. Y. Wang), marui1988.happy@163.com (R. Ma)
*Corresponding author. Email:jianguoyeluo@163.com (J. Y. Wang), marui1988.happy@163.com (R. Ma)
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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 Céa 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 C1 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 methodscontinuous interior penaltypriori error estimateposteriori error analysis

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N15: Error bounds 35J25: Boundary value problems for second-order elliptic equations
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 4 , August 2016 , pp. 517 - 535
Copyright
Copyright © Global-Science Press 2016 

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