Research Interests

My research interests are in numerical methods for differential equations. I work on a number of aspects of spectral collocation methods. Such methods are highly accurate, but often exhibit more severe stability restrictions, lead to full rather than sparse matrices, and are more sensitive to roundoff errors. Many problems require adaptive solution methods, a current topic in my research. For example, to resolve thin boundary layers, it is necessary to adapt the grid or mesh on which the solution is computed to the specific problem solved. Adaptivity becomes even more interesting in time-dependent problems, when the grid has to move as the solution evolves with time. I am also interested in computing invariant manifolds and low-dimensional inertial manifolds for differential equations. These manifolds assist in studying the long-term behaviour of the differential equations involved by effectively reducing the dimension of the problem. This makes bifurcation analysis or long-term simulations feasible. Other research projects are in some numerical aspects of digital signal processing (e.g., Toeplitz systems, wavelets), numerical conformal mapping, and preconditioning and iterative methods for solving large, nonsymmetric systems of linear equations.