Finite difference method for boundary value problem for nonlinear elliptic equation with nonlocal conditions

Abstract

In the paper the convergence of a finite difference scheme for two-dimensional nonlinear elliptic equation in the rectangular domain with the integral boundary condition is considered. The majorant is constructed for the error of the solution of the system of difference equations, and the estimation of this error is obtained. With this aim, the idea of application of the M-matrices for the theoretical investigation of the system of difference equations was developed. Main results for the convergence of the difference schemes are obtained considering the structure of the spectrum and properties of the M-matrices for a wider class of boundary value problems for nonlinear equations with nonlocal conditions. The main advantage of the suggested method is that the error of approximate solution is estimated in the maximum norm.

M-Matrices and Convergence of Finite Difference Scheme for Parabolic Equation With Integral Boundary Condition

“…D is the diagonal matrix with the elements d n ij , prescribed by formula (3.5). For details see [33]. Denote…”

“…Proof. In the article [33] is proved, that the matrix Λ − C under conditions H1 and H2 are M-matrices (see [33] Proof. First of all, it is estimated that the solution z n ij of this system, as i, j = 1, 2, .…”

“…In papers [28,33,38], the theory of M-matrices was used for the investigation of difference systems obtained from the elliptic equation with an integral condition. In many cases, the matrix of the system of difference equations, approximating a differential equation with nonlocal conditions, is characterized by the properties specific to M-matrices.…”

“…In the paper [33], using the properties of M-matrices, the error of the solution of the difference problem for nonlinear elliptic equation with nonlocal boundary condition was estimated. Two aspects of such a method of proof have been noted.…”

“…In this article the methodology of the estimation of the error of the solution as in [33] is applied to a new class of problems with nonlocal conditions, namely, to nonlinear two-dimensional parabolic equations. The main result of the present paper is that the stability and convergence of difference schemes for a parabolic equation with an integral boundary condition in the uniform norm has been proved using the structure of the spectrum of difference operator and the theory of M-matrices.…”