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arXiv · 1204.5798

Convergent filtered schemes for the Monge-Amp\`ere partial differential equation

Brittany D. Froese·Adam M. Oberman
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Abstract

The theory of viscosity solutions has been effective for representing and approximating weak solutions to fully nonlinear Partial Differential Equations (PDEs) such as the elliptic Monge-Amp\`ere equation. The approximation theory of Barles-Souganidis [Barles and Souganidis, Asymptotic Anal., 4 (1999) 271-283] requires that numerical schemes be monotone (or elliptic in the sense of [Oberman, SIAM J. Numer. Anal, 44 (2006) 879-895]. But such schemes have limited accuracy. In this article, we establish a convergence result for nearly monotone schemes. 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-Amp\`ere equation and present computational results on smooth and singular solutions.

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Brittany D. Froese, Adam M. Oberman. 2012-04-26. Convergent filtered schemes for the Monge-Amp\`ere partial differential equation. https://arxiv.org/abs/1204.5798

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