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. Comparedwith 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 Ćea 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.