Superconvergence phenomenon in the finite element method arising from averaging gradients

Abstract

We study a superconvergence phenomenon which can be obtained when solving a 2nd order elliptic problem by the usual linear elements. The averaged gradient is a piecewise linear continuous vector field, the value of which at any nodal point is an average of gradients of linear elements on triangles incident with this nodal point. The convergence rate of the averaged gradient to an exact gradient in theL2-norm can locally be higher even by one than that of the original piecewise constant discrete gradient.