A fourth-order alternating-direction method for difference schemes with nonlocal condition*

Abstract

We consider a finite-difference scheme of fourth-order accuracy for the two-dimensional Poisson equation in a rectangular domain with nonlocal integral conditions in one coordinate direction. The system of finite-difference equations is solved using a generalization of the Peaceman–Rachford alternating-direction implicit method. We prove the convergence of the method and estimate the rate of convergence by using the structure of the spectrum of one-dimensional difference operators with nonlocal integral conditions.