Research

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:

  • Design and analysis of extrapolation methods and acceleration 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.
  • 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.
  • Goal-oriented adaptivity for multiscale methods.

This research is supported by a collaborative grant from the National Science Foundation, DMS 2011519, with Leo Rebholz from Clemson University, and a single PI grant from the NSF, DMS-1852876, 2018-2020 (previously DMS-1719849, 2017-2018). My work on multiscale problems is focused on adaptive enrichment methods, particularly 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.

keywords:
finite element analysis, nonlinear partial differential equations, extrapolation, Anderson acceleration, nonmonotone problems, multiscale methods, discrete comparison principles, uniqueness of discrete solutions, adaptive methods, regularization, goal oriented methods, inverse-kinematics.