Convergent filtered schemes for the Monge-Ampère partial differential equation

Brittany D. Froese, Adam M. Oberman

Research output: Contribution to journalArticlepeer-review

51 Scopus citations

Abstract

The theory of viscosity solutions has been effective for representing and approximating weak solutions to fully nonlinear partial differential equations such as the elliptic Monge-Ampère equation. The approximation theory of Barles and Souganidis[Asymptotic Anal., 4 (1991), pp. 271-283] requires that numerical schemes be monotone (or elliptic in the sense of[A. M. Oberman, SIAM J. Numer. Anal., 44 (2006), pp. 879-895]). But such schemes have limited accuracy. In this article, we establish a convergence result for filtered schemes, which are nearly monotone. This allows us to construct finite difference discretizations of arbitrarily high-order. We demonstrate that the higher accuracy is achieved when solutions are sufficiently smooth. In addition, the filtered scheme provides a natural detection principle for singularities. We employ this framework to construct a formally second-order scheme for the Monge-Ampère equation and present computational results on smooth and singular solutions.

Original languageEnglish (US)
Pages (from-to)423-444
Number of pages22
JournalSIAM Journal on Numerical Analysis
Volume51
Issue number1
DOIs
StatePublished - 2013
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Keywords

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

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