A New Rotated Nonconforming Quadrilateral Element

In this paper, a new nonparametric nonconforming quadrilateral finite element is introduced. This element takes the four edge mean values as the degrees of the freedom and the finite element space is a subspace of P2. Different from the other nonparametric elements, the basis functions of this new element can be expressed explicitly without solving linear systems locally, which can be achieved by introducing a new reference quadrilateral. To evaluate the integration, a class of new quadrature formulae with only three equally weighted points on quadrilateral are constructed. Hence the stiffness matrix can be calculated by the same way with the parametric elements. Numerical results are shown to confirm the optimality of the convergence order for the second order elliptic problems and the Stokes problem.