TY - JOUR T1 - Unified a Priori Error Estimate and a Posteriori Error Estimate of CIP-FEM for Elliptic Equations AU - Wang , Jianye AU - Ma , Rui JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 517 EP - 535 PY - 2018 DA - 2018/05 SN - 8 DO - http://doi.org/10.4208/aamm.2014.m834 UR - https://global-sci.org/intro/article_detail/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 $H^{1+s}$ 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 $C^1$ 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.