Numerical analysis of strongly nonlinear PDEs

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...

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Bibliographic Details
Published in:Acta numerica 2017-05, Vol.26, p.137-303
Main Authors: Neilan, Michael, Salgado, Abner J., Zhang, Wujun
Format: Article
Language:English
Subjects:
Convergence
Finite difference method
Finite element analysis
Finite element method
Mathematics
Methods
Nonlinear analysis
Nonlinear differential equations
Nonlinear equations
Nonlinear programming
Numerical analysis
Numerical methods
Ordinary differential equations
Partial differential equations
Securities prices
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Summary: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.
ISSN:0962-4929
1474-0508
DOI:10.1017/S0962492917000071