Pointwise superconvergence of the gradient for the linear tetrahedral element

We consider the finite element approximation to the solution of a self‐adjoint, second‐order elliptic boundary value problem in three dimensions over a fully uniform mesh of piecewise linear tetrahedral elements. Although the resulting approximation to the gradient is optimal for functions from the approximating space, it is, however, only O ( h ). We show how this can be improved by the recovery , from the finite element solution, of an approximation to the gradient, which is pointwise of a higher order of accuracy than that of the gradient of the finite element approximation. This approximation, termed a recovered gradient function , is, thus, superconvergent . The major task of our analysis is the establishing of an (almost) constant bound on the W   1 1seminorm of the finite element approximation to a smoothed derivative Green's function. © 1994 John Wiley & Sons, Inc.