Partial differential equations of parabolic type are considered. The authors assume that the space domain is discretized by using cell-centered grids (this approach is common in many fields of science and engineering). They also assume that local refinement of the grid is needed in one or more subdomains. If local refinement is needed, then finer grid-cells will be used in the subdomains where  necessary.  Therefore the authors assume that both coarse and fine grid-cells are applied in the discretization of the space domain. Moreover, they assume that not only local refinement in some parts of the space domain is to be used, but also local refinement in the time direction.

This paper studies the stability properties of finite difference schemes that use local refinement in both space and time. The simple case where the problems solved are one-dimensional in space is analyzed first, and the results are then extended to two-dimensional problems. The authors also derive convergence results together with some error estimates. They use energy norms to derive the theoretical results.

The presentation of the results is well organized. Many carefully chosen figures illustrate the cases studied, especially the mixing of coarse and finer grids as well as the organization of time-stepping with different time-steps.

Local refinement is often needed in models arising in scientific and engineering studies. This paper will interest many researchers working on the development of such models.