Pseudo-Conform Polynomial Lagrange Finite Elements on Quadrilaterals and Hexahedra

The aim of this paper is to develop a new class of finite elements on quadrilaterals and hexahedra. The degrees of freedom are the values at the vertices and the approximation is polynomial on each element $K$. In general, with this kind of finite elements, the resolution of second order elliptic problems leads to non-conform approximations.Degrees of freedom are the same than those of isoparametric finite elements. The convergence of the method is analyzed and the theory is confirmed by some numerical results. Note that in the particular case when the finite elements are parallelotopes, the method is conform and coincides with the classical finite elements on structured meshes.