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 language | English (US) |
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Pages (from-to) | 423-444 |
Number of pages | 22 |
Journal | SIAM Journal on Numerical Analysis |
Volume | 51 |
Issue number | 1 |
DOIs | |
State | Published - 2013 |
Externally published | Yes |
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