Year: 2023
Author: Ruo Li, Fanyi Yang
Advances in Applied Mathematics and Mechanics, Vol. 15 (2023), Iss. 5 : pp. 1109–1141
Abstract
We propose a numerical method to solve the Monge-Ampère equation which admits a classical convex solution. The Monge-Ampere equation is reformulated into an equivalent first-order system. We adopt a novel reconstructed discontinuous approximation space which consists of piecewise irrotational polynomials. This space allows us to solve the first-order system in two sequential steps. In the first step, we solve a nonlinear system to obtain the approximation to the gradient. A Newton iteration is adopted to handle the nonlinearity of the system. The approximation to the primitive variable is obtained from the approximate gradient by a trivial least squares finite element method in the second step. Numerical examples in both two and three dimensions are presented to show an optimal convergence rate in accuracy. It is interesting to observe that the approximation solution is piecewise convex. Particularly, with the reconstructed approximation space, the proposed method numerically demonstrates a remarkable robustness. The convergence of the Newton iteration does not rely on the initial values. The dependence of the convergence on the penalty parameter in the discretization is also negligible, in comparison to the classical discontinuous approximation space.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.OA-2022-0047
Advances in Applied Mathematics and Mechanics, Vol. 15 (2023), Iss. 5 : pp. 1109–1141
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 33
Keywords: Monge-Ampère equation least squares method reconstructed discontinuous approximation.
Author Details
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A nonlinear least-squares convexity enforcing 𝐶⁰ interior penalty method for the Monge–Ampère equation on strictly convex smooth planar domains
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https://doi.org/10.1090/cams/39 [Citations: 0]