How to recover the gradient of linear elements on nonuniform triangulations (Q1363447)

It is well known that the theoretical optimal rate of approximation of linear finite elements in \(\mathbb{R}^d\) is of order \(O(h^2)\) in \(L^q\)-norm for \(d<2q\) whereas it is only \(O(h)\) for their gradient. In the present paper, a simple postprocessing is investigated which leads, on completely irregular triangulations, to the \(O(h^2)\) accuracy in \(L^q\)-norm of the interpolation error and \(O(h)\) in \(W^1_q\)-norm provided that the family of triangulations is strongly regular. This postprocessing consists in a simple weighted averaging procedure, which has, moreover, some superconvergence properties on quasiuniform triangulations. The theoretical results are tested on some numerical examples.