People

Tom Lewis

Associate Professor

Office: Petty 144
Email: tllewis3@uncg.edu
Starting year at UNCG: 2013
Office Hours: MW 1:30 -- 2:30 pm and by appointment

Education

Degree(s): Ph.D. in Mathematics, University of Tennessee (2013)

Teaching

Fall 2024
  • MAT-709 LEC (Topics in Computational Math), MWF 12:00-12:50, PETT 217
  • MAT-723 LEC (Numerical Mathematics), MWF 10:00-10:50, PETT 217
  • MAT-799 DTS (Dissertation)
  • MAT-790 IND (Directed Doctoral Research)

Research

Member of the Research Group(s): Applied Math
Current Students: Jesse Paul (Ph.D.), Xiaohuan “Max” Xue (Ph.D.)
Former Students: Kellie Ward (M.A.), Will Stiles (M.A.), Sandamalee Seneviratne (M.A.), Indika Gunawardana (M.A.), Aaron Rapp (Ph.D.)

Research Interests: Numerical Analysis: Numerical Solutions to PDEs, Finite Difference Methods, Finite Element Methods, Discontinuous Galerkin Methods, Fast Solvers for Numerical PDEs. Nonlinear PDEs: Viscosity Solutions for Fully Nonlinear PDEs. Nonlinear Reaction-Diffusion Equations. Application Areas: Deterministic and Stochastic Optimal Control, Optimal Mass Transport, Materials Science, Complex Fluid, Mathematical Biology.

Selected Publications

  • X. Feng, T. Lewis, and K. Ward. A narrow-stencil framework for convergent numerical approxima- tions of fully nonlinear second order PDEs. 2021 UNC Greensboro PDE Conference. Electron. J. Diff. Eqns., Conference 26, p. 59 – 95. 2022.
  • T. Lewis, Q. Morris, and Y. Zhang. Convergence, stability analysis, and solvers for approximating sublinear positone and semipositone boundary value problems using finite difference methods. J. Comput. Appl. Math., Volume 404, 2022.
  • X. Feng, T. Lewis, and A. Rapp. Dual-Wind Discontinuous Galerkin Methods for Stationary Hamilton-Jacobi Equations and Regularized Hamilton-Jacobi Equations. Commun. Appl. Math. Comput. 4, p. 563 – 596, 2022.
  • X. Feng and T. Lewis. A Narrow-stencil finite difference method for approximating viscosity solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Numer. Anal., Volume 59, Number 2, p. 886 – 924, 2021.
  • T. Lewis, A. Rapp, and Y. Zhang. Convergence analysis of symmetric dual-wind discontinuous Galerkin approximation methods for the obstacle problem. J. Math. Anal. and Appl., Volume 485, Issue 2, 22 pages, 2020.
  • X. Feng, T. Lewis, and M. Neilan. Discontinuous Galerkin finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations, J. Comput. Appl. Math., Volume 299, p. 68 — 91. 2016.
  • T. Lewis and M. Neilan. Convergence analysis of a symmetric dual-wind discontinuous Galerkin method, J. Sci. Compute., Volume 59, Issue 3, p. 602 — 625. 2014.
  • X. Feng and T. Lewis. Mixed interior penalty discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic equations in high dimensions, Numer. Methods Partial Differential Equations, Volume 30, Issue 5, p. 1538 — 1557. 2014.
  • X. Feng and T. Lewis. Local discontinuous Galerkin methods for one-dimensional second order fully nonlinear elliptic and parabolic equations, J. Sci. Compute., Volume 59, Issue 2, p. 129 — 157. 2014.
  • X. Feng, C. Kao, and T. Lewis. Convergent finite difference methods for one-dimensional fully nonlinear second order partial differential equations, J. of Comp. and Appl. Math. 254:81-98, 2013.

Brief Biography

Dr. Lewis earned a Ph.D. in 2013 from the University of Tennessee in Knoxville, and he joined the faculty at UNCG the same year. His research focuses on numerical PDEs and applied mathematics.

UNCG Research Spotlight: https://researchmagazine.uncg.edu/spring-2022/awards-add-up/

External Funding

  • Co-PI: Research, Joint with PI Aaron Rapp, “A New Approximation Method for Hamilton-Jacobi Equations,” May 2022 – April 2023, VI NASA EPSCoR. $36,342.06.
  • Former Co-PI: Research, Joint with PI Yi Zhang, “Novel Discontinuous Galerkin Methods for De- terministic and Stochastic Optimization Problems with Inequality Constraints,” September 2021 – August 2024, NSF DMS – Computational Mathematics, Award Number 2111004. $114,944.
  • PI (single): Research, “Narrow-Stencil Numerical Methods for Approximating Nonlinear Elliptic Partial Differential Equations,” July 2021 – June 2024, NSF DMS – Computational Mathematics, Award Number 2111059. $119,972.
  • Co-PI: Conference, “The 35th Southeastern Atlantic Regional Conference on Differential Equations,” 2015, NSF DMS – Applied Mathematics, Award Number 1536101. $24,000.

Honors and Awards

  • College of Arts & Sciences Teaching Excellence Award – Junior Faculty Tenure-Track, 2017 – 2018.
  • Candace Bernard and Robert Glickman Dean’s Professorship in the College of Arts & Sciences, 2016 – 2017.
  • MAA Project NExT (New Experiences in Teaching) Fellow, 2014 – 2015.