We study the convergence of finite difference schemes for approximating elliptic equations of second order with discontinuous coefficients. Two of these finite difference schemes arise from the discretization by the finite element method using bilinear shape functions. We prove an convergence for the gradient, if the solution is locally in H3. Thus, in contrast to the first order convergence for the gradient obtained by the finite element theory we show that the gradient is superclose. From the Bramble–Hilbert Lemma we derive a higher order compact (HOC) difference scheme that gives an approximation error of order four for the gradient. A numerical example is given.
© 2013 by Walter de Gruyter Berlin Boston