Gradient superconvergence on uniform simplicial partitions of polytopes

Superconvergence of the gradient for the linear simplicial finite-element method applied to elliptic equations is a well known feature in one, two, and three space dimensions. In this paper we show that, in fact, there exists an elegant proof of this feature independent of the space dimension. As a result, superconvergence for dimensions four and up is proved simultaneously. The key ingredient will be that we embed the gradients of the continuous piecewise linear functions into a larger space for which we describe an orthonormal basis having some useful symmetry properties. Since gradients and rotations of standard finiteelement functions are in fact the rotation-free and divergence-free elements of Raviart– Thomas and Nédélec spaces in three dimensions, we expect our results to have applications also in those contexts.