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 ${u_h}$ satisfies ${u_h} - u = O({h^r})$. Bramble and Schatz (Math. Comp., v. 31, 1977, pp. 94-111) have constructed, for elements satisfying certain uniformity conditions, a simple function ${K_h}$ such that ${K_h}\; \ast \;{u_h} - u = O({h^{2r - 2}})$ in the interior. Their result is generalized here to obtain similar superconvergent order interior approximations also for derivatives of u.