Research Focus
Finite element analysis, numerical analysis, computational mechanics, applied nonlinear dynamics
My research lies in the interdisciplinary intersection of mathematics, physics, engineering and computational science. More specifically, I do research in computational mathematics that involves numerical analysis and nonlinear dynamics.
Within numerical analysis, I am interested in finite element methods and applied functional analysis. Numerically-stable adaptive high-order finite element methods involving different element shapes and desirable computational properties are of particular interest to me. For example, I have helped develop some of the latest advances in discontinuous Petrov-Galerkin (DPG) methods, which are very effective at computationally solving partial differential equations (PDEs) and which have many physical applications, such as elasticity, viscoelasticity, Stokes flow, acoustics, and electromagnetism.
Within applied nonlinear dynamics, I like to study hydrodynamic stability of classical flows, like plane Couette flow. Here, infinite-dimensional (fluid) dynamical systems are posed in the form of a semidefinite program (SDP) with sum-of-squares (SoS) constraints, and then a computer is used to derive some important stability properties of the dynamical system.
Publications
- High-order polygonal discontinuous Petrov-Galerkin (PolyDPG) methods using ultraweak formulations (with Vaziri Astaneh, A., Mora, J., and Demkowicz, L.), Comput. Methods Appl. Mech. Engrg. (2018), 332:686-711.
- Discrete least-squares finite element methods (with Keith, B., Petrides, S., and Demkowicz, L.), Comput. Methods Appl. Mech. Engrg. (2017), 327:226-255.
- Coupled variational formulations of linear elasticity and the DPG methodology (with Keith, B., Demkowicz, L., and Le Tallec, P.), J. Comput. Phys. (2017), 348:715-731.
- Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application (with Huang, D., Chernyshenko, S., Goulart, P., Lasagna, D., and Tutty, O.), Proc. R. Soc. A (2015), 471(2183).
- Orientation embedded high order shape functions for the exact sequence elements of all shapes (with Keith, B., Demkowicz, L., and Nagaraj, S.) Comput. Math. Appl. (2015), 70(4):353-458.