Higher order local accuracy by averaging in the finite element method

Let u h {u_h} be a Ritz-Galerkin approximation, corresponding to the solution u of an elliptic boundary value problem, which is based on a uniform subdivision in the interior of the domain. In this paper we show that by "averaging" the values of u h {u_h} in the neighborhood of a point x we may (for a wide class of problems) construct an approximation to u ( x ) u(x) which is often a better approximation than u h ( x ) {u_h}(x) itself. The "averaging" operator does not depend on the specific elliptic operator involved and is easily constructed.