A nonconforming Crouzeix-Raviart type finite element on polygonal meshes

A nonconforming lowest order Crouzeix-Raviart type finite element, based on the generalized barycentric coordinates, is constructed on general polygonal (convex or nonconvex) meshes. We reveal a fundamental difference of the Crouzeix-Raviart type degrees of freedom between polygons with odd and even number of vertices, which results in slightly different local constructions of finite elements on these two types of polygons. Because of this, the topological structure of connected regions consisting of polygons with even number of vertices plays an essential role in understanding the global finite element space. To analyze such a topological structure, a new technical tool using the concept of cochain complex and cohomology is developed. Despite the seemingly complicated theoretical analysis, implementation of the element is straightforward. The nonconforming finite element method has optimal a priori error estimates. Proof and supporting numerical results are presented.