Convergent Filtered Schemes for the Monge--Ampère Partial Differential Equation
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Bibliographic record
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.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
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| Insufficient payload (model declined to judge) | 0.008 | 0.000 |
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