I consider myself an engineer and scientist within the area of computational mechanics and applied mathematics. My research is in the intersection engineering, mathematics, physics, and computational science. In particular, I develop finite element methods for challenging problems in engineering science and physics.
My recent research has resulted in the development of a new finite element method, called the Automatic Variationally Stable Finite Element (AVS-FE) method. I am currently working on applying this new method in the modeling of hurricane storm surge, as well as the improvement of existing finite element software libraries for the same models.
Finite Element Methods
My Ph.D. research, supervised by Albert Romkes, was spent focusing on the development of the AVS-FE method. This method is a discontinuous Petrov-Galerkin (DPG) method (developed by Leszek Demkowicz and Jay Gopalakrishnan) making the particular choice of fully continuous trial spaces. As the method is still in its early days of development, the focus has been verification of its properties for a wide range of engineering problems including convection-dominated diffusion, nearly incompressible elastostatics, Cahn-Hilliard equation, and wave propogation to mention a few. After starting my postdoctoral research, I have focused on analysis of the method as well as its application to the modeling of hurricane storm surge. Additionally, as part of a National Science Foundation sponsored project, my work has also included further development of the existing finite element software named discontinuous Galerkin Shallow Water Equations model (DG-SWEM).
The AVS-FE Method
The AVS-FE method is uncondtionally stable, regardless of the underlying differential operator and FE mesh. The stability is derived from the DPG concept of optimal discontinuous test functions computed on-the-fly, whereas the trial space consist of classical FE basis functions.