Constrained Quadrilateral Nonconforming Rotated $Q_1$ Element

Abstract

In this paper, we define a new nonconforming quadrilateral finite element based on the nonconforming rotated $Q$₁ element by enforcing a constraint on each element, which has only three degrees of freedom. We investigate the consistency, approximation, superclose property, discrete Green's function and superconvergence of this element. Moreover, we propose a new postprocessing technique and apply it to this element. It is proved that the postprocessed discrete solution is superconvergent under a mild assumption on the mesh.