Convergence properties in C inside a region for finite difference schemes with a splitting operator as applied to parabolic systems

Abstract

A TWO-LAYER finite-difference scheme with a splitting operator is considered for a general linear system of second-order parabolic equations, and a priori bounds are obtained for its solution in the norms of W 2 2.0 ( Q) and W 2 2 ( Ω 2 ) (defined below). A priori bounds and approximations are obtained below, showing that the finite-difference scheme in question is convergent in C inside a region. The method to be used is that of [1], involving energy inequalities and cut-off functions. The stability and convergence of splitting operator finite-difference schemes were investigated for the.case of general parabolic systems and equations, and various metrics, in [2–5]; in [6] they were studied in the metrics of W 2 2 ( Q) for systems of parabolic equations with no mixed derivatives; in [7], in W 2 2(Ω) , for the first boundary value problem for parabolic equations with no mixed derivatives; and in [8], for the general parabolic equation with data periodic in the space variables.