An Introduction to Numerical Analysis and Approximation Methods for Fully Nonlinear Partial Differential Equations

Dr. Tom Lewis, Associate Professor of Mathematics
Department of Mathematics and Statistics, The University of North Carolina at Greensboro

An Introduction to Numerical Analysis and Approximation Methods for Fully Nonlinear Partial Differential Equations
November 16, 2023 11:00 am

Abstract:  The talk will consist of two parts.  The first part will serve as an introduction to numerical methods with an emphasis on the need for mathematical analysis.  The second part of the talk will focus on motivating monotone methods for approximating solutions to fully nonlinear differential equations.  The definition of a viscosity solution will be introduced using ideas from differential calculus.  We will then focus on how to discretize the key aspects of the definition using finite difference techniques and present the celebrated convergence result of Barles-Souganidis for fully nonlinear differential equations which serves as a generalization of the Lax-Richtmyer equivalence theorem for linear differential equations.  We will end with some current research that seeks to analyze non-monotone schemes.  This talk is accessible to a general audience.