High order local approximations to derivatives in the finite element method

Abstract

Consider the approximation of the solution u of an elliptic boundary value problem by means of a finite element Galerkin method of order r, so that the approximate solution u h {u_h} satisfies u h − u = O ( h r ) {u_h} - u = O({h^r}) . Bramble and Schatz (Math. Comp., v. 31, 1977, pp. 94-111) have constructed, for elements satisfying certain uniformity conditions, a simple function K h {K_h} such that K h ∗ u h − u = O ( h 2 r − 2 ) {K_h}\; \ast \;{u_h} - u = O({h^{2r - 2}}) in the interior. Their result is generalized here to obtain similar superconvergent order interior approximations also for derivatives of u.