My research is focused on the design and analysis of efficient and accurate numerical methods for nonlinear and multiscale partial differential equations. My ongoing work on nonlinear PDE includes:
- Proving comparison theorems and developing uniqueness theory for finite element solutions to nonlinear elliptic equations of nonmonotone type.
- Design of efficient regularized methods for computing solutions to quasilinear PDE, especially those of nonmonotone type.
- Design and analysis of accelerated methods for nonlinear problems.
- Quantifying the behavior of numerical algorithms in the preasymptotic regime.
- Developing global convergence theory for methods that start far from the asymptotic regime.
This research is supported by a single PI grant from the National Science Foundation, DMS-1852876, 2018-2020 (previously DMS-1719849, 2017-2018). My work on multiscale problems is focused on adaptive enrichment methods, particularly goal-oriented adaptivity. Currently, I’m working on:
- Goal-oriented adaptivity for multiscale methods.
- Computation of online basis functions for goal-oriented adaptivity.
I also have a continuing interest in computational geometry, which started from my undergraduate research project. I have worked on developing robust algorithms for inverse-kinematic loop closure, and on characterizing the solution space and singularities of both the geometrical and computational problems.
finite element analysis, nonlinear partial differential equations, Anderson acceleration, nonmonotone problems, multiscale methods, discrete comparison principles, uniqueness of discrete solutions, adaptive methods, regularization, goal oriented methods, inverse-kinematics.