Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation

It is well-known that on uniform meshes the piecewise linear conforming finite element solution of the Poisson equation approximates the interpolant to a higher order than the solution itself. In this paper, this type of superclose property is studied for the canonical interpolant defined by the nodal functionals of several non-conforming finite elements of lowest order. By giving explicit examples we show that some non-conforming finite elements do not admit the superclose property. In particular, we discuss two non-conforming finite elements which satisfy the superclose property. Moreover, applying a postprocessing technique, we can also state a superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself. Finally, we show that an extrapolation technique leads to a further improvement of the accuracy of the finite element solution.