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Mathematics of Computation

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Two-scale method for the Monge-Ampère equation: Convergence to the viscosity solution


Authors: R. H. Nochetto, D. Ntogkas and W. Zhang
Journal: Math. Comp.
MSC (2010): Primary 65N30, 65N12, 65N06, 35J96
DOI: https://doi.org/10.1090/mcom/3353
Published electronically: May 17, 2018
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Abstract: We propose a two-scale finite element method for the Monge-Ampère equation with Dirichlet boundary condition in dimension $ d\ge 2$ and prove that it converges to the viscosity solution uniformly. The method is inspired by a finite difference method of Froese and Oberman, but is defined on unstructured grids and relies on two separate scales: the first one is the mesh size $ h$ and the second one is a larger scale that controls appropriate directions and substitutes the need of a wide-stencil. The main tools for the analysis are a discrete comparison principle and discrete barrier functions that control the behavior of the discrete solution, which is continuous piecewise linear, both close to the boundary and in the interior of the domain.


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Additional Information

R. H. Nochetto
Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
Email: rhn@math.umd.edu

D. Ntogkas
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: dimitnt@gmail.com

W. Zhang
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854
Email: wujun@math.rutgers.edu

DOI: https://doi.org/10.1090/mcom/3353
Keywords: Monge-Amp\`ere equation, viscosity solution, two-scale method, monotone scheme, convergence, regularization
Received by editor(s): June 19, 2017
Received by editor(s) in revised form: December 10, 2017, December 11, 2017, and December 19, 2017
Published electronically: May 17, 2018
Additional Notes: The first author was partially supported by the NSF Grant DMS -1411808, the Institut Henri Poincaré (Paris), and the Hausdorff Institute (Bonn).
The second author was partially supported by the NSF Grant DMS -1411808 and the 2016-2017 Patrick and Marguerite Sung Fellowship of the University of Maryland.
The third author was partially supported by the NSF Grant DMS -1411808 and the Brin Postdoctoral Fellowship of the University of Maryland.
Article copyright: © Copyright 2018 American Mathematical Society

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