On the uniform convergence of certain difference schemes

Abstract

IN THIS paper we obtain uniform evaluations of the speed of convergence of some difference schemes for which the maximum principle has not been established. Such are: 1. (1) schemes for increasing the order of accuracy on a rectangular network for Poisson's equation in the case of two and three ( p = 2, 3) variables, considered in [1]; 2. (2) schemes for increasing the order of accuracy with a decomposing operator for the heat conduction equation with constant coefficients for p = 2,3, considered in [2]; 3. (3) schemes with a decomposing operator for a parabolic equation with variable coefficients [3]. In the investigation the method of energy inequalities is used, by means of which evaluations of the solution in the norm of the space W 2 0(2)are established. From the a priori evaluations obtained the uniform convergence follows on the basis of the difference analogue of the theorem of embedding which has been proved. The method, which we use to obtain the a priori evaluation, was used in [4] to prove the uniform convergence of a difference scheme which approximates the Dirichlet problem for a second order differential equation of elliptic type with a mixed derivative. From there we also borrowed a method of proof of the difference theorem of embedding.