Recent Developments in Numerical Methods for Fully Nonlinear Second Order Partial Differential Equations

This article surveys the recent developments in computational methods for second order fully nonlinear partial differential equations (PDEs), a relatively new subarea within numerical PDEs. Due to their ever increasing importance in mathematics itself (e.g., differential geometry and PDEs) and in ma...

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Bibliographic Details
Published inSIAM review Vol. 55; no. 2; pp. 205 - 267
Main Authors Feng, Xiaobing, Glowinski, Roland, Neilan, Michael
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
Subjects
Applied mathematics
Astrophysics
Communities
Computer science
Engineering
Estimates
Geometry
Image processing
Lagrange multiplier
Mathematical models
Meteorology
Nonlinear differential equations
Nonlinear programming
Nonlinearity
Numerical analysis
Partial differential equations
Studies
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Summary:This article surveys the recent developments in computational methods for second order fully nonlinear partial differential equations (PDEs), a relatively new subarea within numerical PDEs. Due to their ever increasing importance in mathematics itself (e.g., differential geometry and PDEs) and in many scientific and engineering fields (e.g., astrophysics, geostrophic fluid dynamics, grid generation, image processing, optimal transport, meteorology, mathematical finance, and optimal control), numerical solutions to fully nonlinear second order PDEs have garnered a great deal of interest from the numerical PDE and scientific communities. Significant progress has been made for this class of problems in the past few years, but many problems still remain open. This article intends to introduce these current advancements and new results to the SIAM community and generate more interest in numerical methods for fully nonlinear PDEs.
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ISSN:0036-1445
1095-7200
DOI:10.1137/110825960