Numerical solution of the nonlocal boundary value problem for elliptic equations

In the present paper a second order of accuracy two-step difference scheme for an approximate solution of the nonlocal boundary value problem for the elliptic differential equation -v ''(t) + Av(t) = f(t) (0 <= t <= T); v(0) = v(T) + phi, integral(T)(0) v(s)ds = psi in an arbitrary Banach space E with the strongly positive operator A is presented. The stability of this difference scheme is established. In application, the stability estimates for the solution of the difference scheme for the elliptic differential problem with the Neumann boundary condition are obtained. Additionally, the illustrative numerical result is provided.