Study of the rate of convergence of variational difference schemes for second-order elliptic equations in a two-dimensional field with a smooth boundary

Abstract

IN this paper the rate of convergence of variational-difference schemes (VDS) is studied when the differential properties of the solution are varied, the estimates of the convergence rate being derived in different norms. The differential properties of the solution are determined by the properties of the coefficients and of the right side of the equation (we consider the boundary of the domain to be sufficiently smooth). We will vary the properties of the right side of the equation only, and with respect to the coefficients we will suppose that they have the smoothness which is required in the proofs. In obtaining the estimates we have attempted only to explain the order in h ( h is the step of the net) of the approximation to zero of the error, therefore we will not deduce in detail the values of the constants in the estimates. Estimates of the rate of convergence will be obtained in terms of the data of the problem. In the majority of cases the estimates obtained are exact with respect to order in the sense of section 4 of [1]. Section 1 contains auxiliary propositions. In section 2 a new estimate of the error in the norm of the space L 2 is obtained, and then using theorems of V. P. Il'in in [2, 3] a scale of error estimates is obtained for the right side from L 2. Section 3 is devoted to error estimates in net norms. As a corollary of the estimates obtained we derive estimates of the error in the norm of the space C. Almost all the results are derived for both the first and third boundary value problems. In the present paper we have considered only the case of a two-dimensional domain, but it seems to us that the transfer of all the estimates in the norms of Hilbert spaces to the multidimensional case taking the geometrical constructions into account [4], presents no difficulty.