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Recent Developments in Numerical Methods for Fully Nonlinear Second Order Partial Differential Equations
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DOI：10.1137/110825960.png)
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Abstract
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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.
Keywords:
fully nonlinear PDEs
Monge-Ampere-type equations
viscosity solutions
moment solutions
vanishing moment method
finite element methods
spectral methods
discontinuous Galerkin methods
finite difference methods
augmented Lagrangian methods
least squares methods
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