New nonconforming finite elements on arbitrary convex quadrilateral meshes

In this paper, we construct new nonconforming finite elements on the meshes consisting of arbitrary convex quadrilaterals, especially for the quadratic and cubic cases. For each case, we first define a quadrilateral element that adopts edge moments as the degrees of freedom (DoFs), and then enforce a linear constraint on this element. We have, for the quadratic case, eight degrees of freedom per element and, for the cubic case, eleven DoFs per element, respectively. The dimensions and the bases of different types for the global finite element spaces are provided. We consider the approximations of two-dimensional second order elliptic problems for both of these elements. Error estimates with optimal convergence order in both broken H1H1 norm and L2L2 norm are given. Moreover, we consider the discretization of the Stokes equations adopting our quadratic element to approximate each component of the velocity, along with piecewise discontinuous P1P1 element for the pressure. This mixed scheme is stable and optimal error estimates both for the velocity and the pressure are also achieved. Numerical examples verify our theoretical analysis.