Solutions to singularly perturbed problems may rapidly change and develop layers. Therefore, the numerical approximation and the representation of the solution is a rather difficult undertaking. Special techniques as upwinding, streamline diffusion, or exponential fitting are required. Another way consists in using a special grid structure: Near layers, the grid shall be condensed.
In 1990, G. I. Shishkin, one of the authors of the paper under review, could prove uniform convergence of almost optimal order (up to a small logarithmic factor) of a scheme that is not exponentially fitted: He proposed a piecewise equidistant mesh with one half of the grid points lying in the layer region. This was a break through in the numerical analysis for singularly perturbed problems.
In the present paper, the authors consider singularly perturbed parabolic problems with Dirichlet boundary. Since the error depends on the perturbation parameter \(\varepsilon\), so-called \(\varepsilon\)-uniformly convergent schemes have to be constructed. The authors focus on the time discretization and develop difference schemes with arbitrary order of convergence in time for smooth solutions. So, earlier work, in which second-order in space but first-order in time schemes has been proposed, can be essentially improved by means of the defect correction method. Numerical tests illustrate the method as well as the second-order accuracy in space.