Necessary conditions for ɛ-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers

Abstract

A grid approximation of the boundary value problem for a singularly perturbed parabolic reaction-diffusion equation is considered in a domain with the boundaries moving along the axis x in the positive direction. For small values of the parameter ɛ (this is the coefficient of the higher order derivatives of the equation, ɛ ∈ (0, 1]), a moving boundary layer appears in a neighborhood of the left lateral boundary S1L. In the case of stationary boundary layers, the classical finite difference schemes on piece-wise uniform grids condensing in the layers converge ɛ-uniformly at a rate of O(N−1lnN + N0), where N and N0 define the number of mesh points in x and t. For the problem examined in this paper, the classical finite difference schemes based on uniform grids converge only under the condition N−1 + N0−1 ≪ ɛ. It turns out that, in the class of difference schemes on rectangular grids that are condensed in a neighborhood of S1L with respect to x and t, the convergence under the condition N−1 + N0−1 ≤ ɛ1/2 cannot be achieved. Examination of widths that are similar to Kolmogorov’s widths makes it possible to establish necessary and sufficient conditions for the ɛ-uniform convergence of approximations of the solution to the boundary value problem. These conditions are used to design a scheme that converges ɛ-uniformly at a rate of O(N−1lnN + N0).