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
C´ea 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
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.