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Numerical analysis of strongly nonlinear PDEs*

Published online by Cambridge University Press:  05 May 2017

Michael Neilan ,
Abner J. Salgado  and
Wujun Zhang
Michael Neilan
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA E-mail: neilan@pitt.edu
Abner J. Salgado
Affiliation:
Department of Mathematics, The University of Tennessee, Knoxville, TN, USA E-mail: asalgad1@utk.edu
Wujun Zhang
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ, USA E-mail: wujun@math.rutgers.edu
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Abstract

We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and non-convex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental result in this area which states that stable, consistent and monotone schemes converge as the discretization parameter tends to zero. We review methodologies to construct finite difference, finite element and semi-Lagrangian schemes that satisfy these criteria, and, in addition, discuss some rather novel tools that have paved the way to derive rates of convergence within this framework.

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Type
Research Article
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Acta Numerica , Volume 26 , 01 May 2017 , pp. 137 - 303
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© Cambridge University Press, 2017 

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