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

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 lowerH1+sweak 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 auxiliaryC1finite 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.