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Theses

Monotone finite difference discretization of degenerate elliptic partial differential equations using Voronoi's first reduction

Abstract : In this thesis, we show how Voronoi's first reduction may be used in order to build monotone finite difference discretizations on Cartesian grids of some degenerate elliptic differential operators. We recommend a specific, second-order consistent discretization of two- and three-dimensional linear anisotropic differential operators involving both a first- and a second-order term. We prove the quasi-optimality of this construction. We study some properties on the regularity and the compactness of Voronoi's first reduction in dimension four. We design a method allowing to efficiently approximate Randers distances and associated optimal transport distances, using a large deviations principle. We discretize the Pucci and Monge-Ampère operators. The resulting discretizations are written as maxima of discrete operators; in dimension two, we show that these maxima admit closed-form formulae, reducing the numerical cost of their evaluation. We study the well-posedness, and in some cases the convergence, of a numerical scheme for the second boundary value problem for the Monge-Ampère equation. We present a numerical application to the far-field refractor problem in nonimaging optics.
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https://tel.archives-ouvertes.fr/tel-03485421
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Submitted on : Friday, December 17, 2021 - 11:43:09 AM
Last modification on : Wednesday, April 20, 2022 - 3:44:07 AM
Long-term archiving on: : Friday, March 18, 2022 - 6:52:21 PM

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Guillaume Bonnet. Monotone finite difference discretization of degenerate elliptic partial differential equations using Voronoi's first reduction. Numerical Analysis [math.NA]. Université Paris-Saclay, 2021. English. ⟨NNT : 2021UPASM042⟩. ⟨tel-03485421⟩

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