Finite-difference methods for solving a nonlocal boundary value problem for a multidimensional parabolic equation with boundary conditions of integral form

Abstract

The article considers a non-local boundary value problem for a multidimensional parabolic equation with integral boundary conditions. To solve the problem, we obtain an a priori estimate in differential form, which implies the uniqueness and stability of the solution with respect to the right-hand side and initial data on the layer in the $L_2$-norm. For the numerical solution of a nonlocal boundary value problem, a locally one-dimensional (economical) difference scheme by A.A. Samarskii with the order of approximation $O(h^2+\tau)$, the main idea of which is to reduce the transition from layer to layer to the sequential solution of a number of one-dimensional problems in each of the coordinate directions. Using the method of energy inequalities, a priori estimates are obtained, which imply uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem in the $L_2$-norm at a rate equal to the order of approximation of the difference scheme. An algorithm for the numerical solution is constructed.