On a difference scheme of fourth order of accuracy for the Bitsadze–Samarskii type nonlocal boundary value problem

Abstract

The Bitsadze–Samarskii type nonlocal boundary value problem math image for the differential equation in a Hilbert space H with the self‐adjoint positive definite operator A with a closed domain D(A) ⊂ H is considered. Here, f(t) be a given abstract continuous function defined on [0,1] with values in H, φ and ψ be the elements of D(A), and λj are the numbers from the set [0,1]. The well‐posedness of the problem in Hölder spaces with a weight is established. The coercivity inequalities for the solution of the nonlocal boundary value problem for elliptic equations are obtained. The fourth order of accuracy difference scheme for approximate solution of the problem is presented. The well‐posedness of this difference scheme in difference analogue of Hölder spaces is established. For applications, the stability, the almost coercivity, and the coercivity estimates for the solutions of difference schemes for elliptic equations are obtained. Mathematical Methods in the Applied Sciences. Copyright © 2012 John Wiley & Sons, Ltd.