Unified a Priori Error Estimate and a Posteriori Error Estimate of CIP-FEM for Elliptic Equations (Advances in Applied Mathematics and Mechanics, 2018)
This paper is devoted to a unified a priori and a posteriori error analysis ofCIP-FEM (continuous interior penalty finite element method) for second-order ellipticproblems. Compared with the classic a priori error analysis in literature, our techniquecan easily apply for any type regularity assumption on the exact solution, especiallyfor 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 andCéa Lemma for conforming finite element methods can not be applied immediatelywhen 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 utilizedin the explicit a posteriori error analysis of CIP-FEM.