Finite Difference Scheme for a Singularly Perturbed Parabolic Equations in the Presence of Initial and Boundary Layers

Abstract

The grid approximation of an initial-boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation. The secondorder spatial derivative and the temporal derivative in the differential equation are multiplied by parameters ε 2 1 and ε 2 2 , respectively, that take arbitrary values in the open-closed interval (0, 1]. The solutions of such parabolic problems typically have boundary, initial layers and/or initial-boundary layers. A priori estimates are constructed for the regular and singular components of the solution. Using such estimates and the condensing mesh technique for a tensor-product grid, piecewise-uniform in x and t, a difference scheme is constructed that converges ε-uniformly at the rate O(N −2 ln 2 N + N −1 0 ln N0), where (N + 1) and (N0 + 1) are the numbers of mesh points in x and t respectively.