Pseudo-conforming polynomial finite elements on quadrilaterals

The aim of this paper is to present a new class of finite elements on quadrilaterals where the approximation is polynomial on each element K. In the case of Lagrange finite elements, the degrees of freedom are the values at the vertices and in the case of mixed finite elements the degrees of freedom are the mean values of the fluxes on each side. The degres of freedom are the same as those of classical finite elements. However, in general, with this kind of finite elements, the resolution of second order elliptic problems leads to non conforming approximations. In the particular case when the finite elements are parallelograms, we can notice that our method is conform and coincides with the classical finite elements on structured meshes. First, a motivation for the study of the Pseudo-conforming polynomial finite elements method is given, and the convergence of the method established. Then, numerical results that confirm the error estimates, predicted by the theory, are presented.