The accuracy of the solution of a difference boundary value problem for a fourth-order elliptic operator with mixed boundary conditions

Abstract

An estimate of the accuracy of a conservative difference scheme in an energy net metric is established, when the solution of the initial problem belongs to Sobolev spaces. The accuracy of difference schemes for fourth-order elliptic differential operators when free-boundary type boundary conditions are imposed were investigated in [1–4] assuming that the solution and the coefficients belong to the spaces C (4)(Ω) and C (4)(Ω) respectively. A new approach, proposed in [5], when investigating the accuracy of difference schemes uses special averaging operators and enables one to obtain an estimate of the accuracy of the net method when there are requirements regarding the smoothness of the coefficients and the solution of the problem in terms of Sobolev spaces. Estimates of the accuracy of these schemes for certain problems with fourth-order elliptic operators were obtained in [6–8]. In this paper we investigate the accuracy of a difference scheme from [4] assuming that the solution and the coefficients of the initial problem belong to the spaces W 2 (k)(Ω) , kϵ [3, 4] and W 2 (m)(Ω) , mϵ (1, 2) respectively.