Grid approximation of singularly perturbed parabolic reaction-diffusion equations with piecewise smooth initial-boundary conditions

A Dirichlet problem is considered for a singularly perturbed parabolic reaction-diffusion equation with piecewise smooth initial-boundary conditions on a rectangular domain. The higher-order derivative in the equation is multiplied by a parameter epsilon(2); epsilon is an element of (0, 1]. For small values of epsilon, a boundary and an interior layer arises, respectively, in a neighbourhood of the lateral part of the boundary and in a neighbourhood of the characteristic of the reduced equation passing through the point of nonsmoothness of the initial function. Using the method of special grids condensing either in a neighbourhood of the boundary layer or in neighbourhoods of the boundary and interior layers, special epsilon-uniformly convergent difference schemes axe constructed and investigated. It is shown that the convergence rate of the schemes crucially depends on the type of nonsmoothness in the initial-boundary conditions.