Finite difference scheme for a singularly perturbed parabolic equations in the presence of initial and boundary layers

The grid approximation of an initial-boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation. The second-order spatial derivative and the temporal derivative in the differential equation are multiplied by parameters eε^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 ε̄-uniformly at the rate 0(N-² In² N + N _o^-1 In N_o), where (N + 1) and (N₀ + 1) are the numbers of mesh points in x and t respectively.