NDLI: Survey and Review (SIREAD Vol. 55 Issue. 2)

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Survey and Review (SIREAD Vol. 55 Issue. 2)

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Higham, Desmond J.
Copyright Year	2013
Abstract	The Survey and Review article in this issue, Recent Developments in Numerical Methods for Fully Nonlinear Second Order Partial Differential Equations, by Xiao-bing Feng, Roland Glowinski, and Michael Neilan, reviews recent progress in numerical methods for second order PDEs that are nonlinear in the highest order derivatives of the solution: so-called fully nonlinear PDEs. Perhaps the best known examples from this very general class are the Monge--Ampère equations, which arise in differential geometry and optimal transport. In this article the authors also point out applications of fully nonlinear PDEs in image registration, cosmology, weather forecasting, fluid dynamics, mesh generation, and stochastic control, and give many examples of challenging nonlinearities. They make it clear that progress in the design and analysis of numerical methods often goes hand in hand with progress in the underlying PDE theory, and insights can be passed in either direction. This theme was also explored in the (much more linear) review of the finite element method in the recent historically oriented Survey and Review article From Euler, Ritz, and Galerkin to Modern Computing, by Martin Gander and Gerhard Wanner, SIAM Review, 54-4 (2012), pp. 627--666. A theory for classical solutions to fully nonlinear PDEs was well-developed by the mid-1980s. However, when such solutions don't exist, it is typically not possible to construct an alternative weak solution theory with the familiar integration by parts trick. Instead, the nonvariational concept of viscosity solutions has been developed. As the authors explain, this differentiation by parts approach does not lend itself readily to the design of numerical methods. In the main body of the article, the authors discuss recent developments for two main classes of numerical method: finite difference methods and Galerkin-type methods. For each class, they review the state of the art with respect to theoretical understanding, and, with the aim of stimulating fresh ideas, highlight open questions and numerical challenges. They give illustrative computational examples on a range of problems and discuss implementation issues, including the solution of the resulting nonlinear algebraic systems in a setting where nonuniqueness is often a feature of the underlying problem. Readers with interests in PDEs and/or numerical analysis will find that this timely and informative article sheds light on a challenging, active research frontier.
Starting Page	203
Ending Page	203
Page Count	1
File Format	PDF
ISSN	00361445
DOI	10.1137/130973375
e-ISSN	10957200
Journal	SIAM Review (SIREAD)
Issue Number	2
Volume Number	55
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2013-01-01
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Computational Mathematics

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