My research field is numerical analysis (click here for a non-technical explanation of what this is) – specifically, the numerical analysis of partial differential equations. I am also interested in finding new ways to promote mathematics in adult and K-12 populations, and intersections between mathematics and the humanities. (See Outreach Activities for more on these latter topics.)
Some specific numerical analysis research problems/areas I have worked on are as follows:
- Finite element methods for elliptic problems, in particular the p version and hp version.
(What are finite elements? Click here for a non-technical explanation)
- Numerical approximation of thin domains such as plates and shells.
- Buckling of thin plates. (An explanation without mathematical formulas can be found here: Research on Buckling).
- Computational analysis of woven composite materials.
- Computational analysis of domains with small holes (ongoing).
The p and hp versions of the finite element method represent a very successful collaboration between mathematicians like Ivo Babuška and his collaborators, and engineers like Barna Szabo and his team. The commercial engineering design program Stress Check (ESRD, St. Louis) incorporates several mathematical techniques developed under such collaboration. I am proud to have participated in this research. (Click here for an example of how math and engineering interact.)
I have been on the editorial boards of SIAM Journal on Numerical Analysis (1999-2004) and Journal of Computational and Applied Mathematics (1993-Present). I have directed five doctoral dissertations. My research has been supported by the National Science Foundation and the Air Force Office of Scientific Research (AFOSR). I have been participating in the Finite Element Circus for many years. We have our own Circus Repository of Poems, poking through which may even turn up a poem penned by me.