Numerical solution of non-linear non-local problems for elliptic equations

Abstract

A non-local problem for an elliptic equation in a rectangular domain was investigated. A rectangular grid for the corresponding difference problem was constructed and the error of the approximate solutions of non-local problems was estimated. Various application problems (heat conductivity [1, 2, 3], fluid mechanics [4], the theory of elasticity and shells [5], etc.) are reduced to non-local boundary value problems. Non-local boundary conditions are especially difficult for justification of classical finite difference schemes due to the complexity of the structure of the matrices obtained from systems of equations. This difficulty manifests itself especially in the justification of numerical methods in the case of non-linear equations. In this paper we consider the non-local boundary value problem for a quasi-linear equation. We found the numerical solutions of stated problem using the finite difference method, and estimated the error of the approximate solutions of non-local problems.