A new technique to study special difference schemes numerically for a Dirichlet problem
on a rectangular domain (in x, t) is considered for a singularly perturbed parabolic reaction-diffusion
equation with a perturbation parameter ε ; ε ∈ (0, 1]. A well known difference scheme on
a piecewise-uniform grid is used to solve the problem. Such a scheme converges ε-uniformly in the
maximum norm at the rate O(N^{-2} ln^2 N + N^{-1}_0) as N, N_0 → ∞, where N + 1 and N_0 + 1
are the numbers of nodes in the spatial and time meshes, respectively; for ε ≥ m ln^{-1} N the
scheme converges at the rate O(N^{-2} + N^{-1}_0). In this paper we elaborate a new approach based on the
consideration of regularized errors in discrete solutions, i.e., total errors (with respect to both
variables x and t), and also fractional errors (in x and in t) generated in the approximation
of differential derivatives by grid derivatives. The regularized total errors agree well with known
theoretical estimates for actual errors and their convergence rate orders. It is also shown that a
“standard” approach based on the “fine grid technique” turns out inefficient for numerical study
of difference schemes because this technique brings to large errors already when estimating the
total actual error.
