The main thrust of my research is the construction, analysis, and evaluation of numerical methods for partial differential equations (PDE's), but I also have had interests in related areas such as the construction of mathematical models for physical and biological systems.
Approximate solution of PDE's is frequently computationally expensive, even for problems that are conceptually simple. I have been studying ways of using adaptivity to make some of these calculations more efficient and robust. For time-dependent problems the use of meshes that move smoothly with time can be of significant value in producing high quality solutions to difficult problems. Although a general solution to the question of how to use such meshes has not yet been found, there are many situations that I have looked at with my students in which such procedures can be both effective and simple.
The computation of free surface flows is important in several of the projects that I am working on at the moment. These involve the formation of drops under various conditions, modeling of the flow of a fluid over a solid surface, and two fluid flows.