My personal interests in this broad area of numerical analysis are concerned mainly with local refinement techniques for transient boundary and initial value problems.

Parabolic partial differential equations are used to model a variety of time-dependent diffusive or convective-diffusive processes. When the diffusion dominates the convection, the solutions of standard discretizations tend to be fairly stable. On the other hand, when the convective properties govern the process, upstream weighting techniques combined with explicit time-stepping allow efficient resolution of the hyperbolic properties of the solution.

The solution of these equations may possess highly localized properties both in space and in time. In many physical applications, these properties are due to stationary features such as wells, cracks, obstacles, domain boundaries, etc., which are fixed in space. In many other cases they are moving in time -- moving point loads, sharp fronts, etc. Due to the size of many applications, the local properties of the solution cannot be resolved using uniform grids even with the largest of today's supercomputers. Adaptive local grid refinement techniques are an attractive alternative for resolving the localized characteristics within a given error tolerance while saving computational resources.

In my work I have studied theoretically the stability and the convergence of implicit finite difference approximations with variable time steps in space. These approximations can be characterized in the following way: first one introduces a global time discretization for the whole domain; next, in some subdomains, chosen using some adaptive mesh refinement or some a priori information on potential rapid local temporal change of the solution, one introduces time steps that are fractions of the global time step. In this way a composite time-space mesh is introduced. One of the most important problems that arises in such situations is the construction of a stable and accurate approximation of the time-dependent problem. Difficulties arise at the interface between the subregions with different time steps. The available literature shows that these are the places that govern the stability and the accuracy of the whole scheme to a great extent. Based on the interfacial treatment, the advantages and disadvantages of the different approaches could be clearly assessed.

One of my recent papers on this subject, Finite Difference Scheme on Composite Grids with Refinement in Time and Space. (SIAM J.Num.Anal. 31 (94), pp. 1605-1622.), offers a comprehensive study of some the mentioned above problems. This electronic copy does not include some of the plots used in the original paper, so please refer to the publication in the journal or e-mail me if you need them.