Locally One-Dimensional Difference Scheme for the Third Boundary Value Problem for a Parabolic Equation of the General Form with a Nonlocal Source

Abstract

We consider a locally one-dimensional scheme for an equation of parabolic type of the general form in a p-dimensional parallelepiped, obtain an a priori estimate for its solution, and prove that the solutions of this scheme converge to a solution of the equation at the rate O(|h|² + τ), where |h|² = h₁² + · · · + h_p² and p_α, α = 1,..., p, and τ are the steps in the space and time variables. We do not assume that the operator in the leading part of the equation is sign definite.