Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher

Brittany D. Froese, Adam M. Oberman

Research output: Contribution to journalArticlepeer-review

60 Scopus citations

Abstract

The elliptic Monge-Ampère equation is a fully n onlinear partial differential equation that originated in geometric surface theory and has been applied in dynamic meteorology, elasticity, geometric optics, image processing, and image registration. Solutions can be singular, in which case standard numerical approaches fail. Novel solution methods are required for stability and convergence to the weak (viscosity) solution. In this article we build a wide stencil finite difference discretization for the Monge-Ampère equation. The scheme is monotone, so the Barles-Souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation. Solutions of the scheme are found using a damped Newton's method. We prove convergence of Newton's method and provide a systematic method to determine a starting point for the Newton iteration. Computational results are presented in two and three dimensions, which demonstrates the speed and accuracy of the method on a number of exact solutions, which range in regularity from smooth to nondifferentiable.

Original languageEnglish (US)
Pages (from-to)1692-1714
Number of pages23
JournalSIAM Journal on Numerical Analysis
Volume49
Issue number4
DOIs
StatePublished - 2011
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Keywords

  • Convexity constraints
  • Fully nonlinear elliptic partial differential equations
  • Monge-Ampère equations
  • Monotone schemes
  • Nonlinear finite difference methods
  • Viscosity solutions

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