About

Finite Element Exterior Calculus

Recently, theoretical tools from differential geometry have been incorporated in the development of finite element methods. The resulting framework, known as finite element exterior calculus (FEEC), enables design and analysis of finite element methods such that certain topological and geometric structures of partial differential equations are preserved. In particular, we are interested in using FEEC to develop structure-preserving finite element methods for fluid equations (e.g., Stokes flow).

Efficient discretizations of fractional differential equations

I am developing and analyzing efficient numerical methods for solving fractional differential equations (FDEs). FDEs are nonlocal integro-differential equations for which traditional approximation techniques require significant computational complexity, hence new approaches are needed. These types of equations arise in models of complex materials, anomalous transport, ultrasound, and other applications involving nonlocality and memory effects.

Boundary conditions for fractional diffusion operators

As FDEs involve nonlocal operators, the concept of boundary conditions must be reinterpreted in this context. I am currently working on developing numerical methods for dealing with nonlocal reflecting boundary conditions in anomalous diffusion problems.

Presentations and Posters

  • Boundary conditions for tempered fractional diffusion
    Invited Presentation at SIAM CSE 2019, Minisymposium on Theoretical and Computational Aspects in Nonlocal and Material Science Modeling, in Spokane, WA, February 2019.
  • What Is the Fractional Laplacian?
    Invited Presentation at ICOSAHOM 2018 in London, UK, July 2018.
  • What Is the Fractional Laplacian?
    Poster Presentation at ICERM Workshop on Fractional PDEs: Theory, Algorithms and Applications in Providence, RI, June 2018.
  • What Is the Fractional Laplacian?
    Invited Poster Presentation at MANNA: Modeling, Analysis and Numerics for Nonlocal Applications, Santa Fe, NM, Dec 2017.
  • Generalized Petrov-Galerkin Schemes of Linear Complexity for Distributed Order Initial Value Problems
    Invited Presentation at SIAM Annual Meeting, Pittsburgh, PA, July 2017.
  • A Tunably-Accurate Spectral Method with Linear Complexity for Multi-Term Fractional Differential Equations on the Half Line
    Poster Presentation at SIAM CSE, Atlanta, GA, Feb 2017.
  • A Tunably-Accurate Spectral Method with Linear Complexity for Multi-Term Fractional Differential Equations on the Half Line
    Invited Poster Presentation at A Workshop on Future Directions in Fractional Calculus Research and Applications, MSU, Oct 2016.
  • Spectral methods for fractional differential equations on the half line with tunable accuracy
    Invited Presentation at The 2016 SIAM Annual Meeting, Boston, MA, July 2016.
  • Efficient and tunably accurate spectral methods for fractional differential equations on the half line
    Invited Presentation at Minisymposium on Recent Advances in Computational PDEs and their Applications, AIMS 2016, Orlando, FL, July 2016.
  • Efficient and tunably accurate Laguerre Petrov-Galerkin spectral methods for fractional differential equations on the half line
    Invited Presentation at Minisymposium on New Trends in Computational Methods for Fractional PDEs: Theory, Numerics, and Applications, ICOSAHOM 2016, Rio de Janeiro, Brazil
  • Asymptotic-preserving space-time discontinuous Galerkin methods for a class of relaxation systems
    Contributed Presentation at SIAM CSE, March 2016.

Professional Affiliations

  • SIAM - Graduate Student Member, Brown University Student Chapter Executive Committee Member
  • AWM - Graduate Student Member, Brown University Student Chapter President
  • Rose Whelan Society - Society for Women in Math at Brown University
  • AMS - Graduate student member
  • Phi Kappa Phi Honor Society