Compatible convergence estimates in the method of refinement by higher-order differences

We consider the Dirichlet problem for an elliptic equation with constant coefficients, which is solved by a difference scheme of second-order accuracy. By using the approximate solution, we correct the right-hand side of the difference scheme. We show that the solution of the corrected scheme is convergent at the rate O (| h | m ) in the discrete L 2 -norm provided that the solution of the original problem belongs to the Sobolev space with exponent m ∈ [2, 4].