My current work at UNC involves fluid-structure interaction in biological flows; in particular, I am interested in numerical methods for simulating cardiac flow. These problems involve complex geometries, multiple physics scales (e.g., blood flow and clotting), and high requirements on Navier-Stokes solvers (such as turbulence modeling and schemes that preserve mass and momentum discretely).
Most of the work I did for my PhD thesis (which I am still building upon) involved the idea of reduced order models: that is, one can replace a large, computationally expensive model with a low rank surrogate. In particular, I am interested in how one can stabilize reduced order models to improve accuracy, and, in some circumstances, fix convergence issues.
Finite Element Methods
During my PhD I worked on a few projects with C1 finite elements: the work was novel because we obtained optimal error estimates for high-order finite elements for the Quasi-Geostrophic Equation (QGE). We also wrote a general implementation, for unstructured triangulations, that can calculate shape functions and their derivatives on physical elements by use of a non-affine mapping.
At RPI I continued this work and did some fundamental research on superconvergence of function derivatives. This project is motivated by the use of compatibility boundary conditions, or boundary conditions based on the partial differential equation itself, in finite difference methods. Our work in applying these finite difference techniques in a finite element context is ongoing.
A common theme in my finite element research is my interest in higher-order methods: I am interested both in how to use higher degree polynomial approximations (such as the Argyris element) and also how to prove superconvergence results.
A large part of my research involves scientific software: I am a chief developer of both deal.II and IBAMR. In both cases I am interested in topics like parallelization and higher-order approximation of geometries.