A new family of nonconforming finite elements on quadrilaterals

In this paper, we generalize one first order nonconforming quadrilateral finite element proposed by Lin, Tobiska and Zhou to any odd order. We construct degrees of freedom for shape function spaces for this family of elements and show their unisolvency. In addition, we present a medium a priori error analysis for this family of nonconforming elements on general quadrilateral meshes. Compared with the classical error analysis of the nonconforming finite element method, the a priori analysis herein only needs the H1 regularity of the exact solution. Numerics are presented to demonstrate theoretical results which in particular show dependence of convergence on mesh distortion parameters α.