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Second order monotone finite differences discretization of linear anisotropic differential operators

Authors: J. Frédéric Bonnans, Guillaume Bonnet and Jean-Marie Mirebeau
Journal: Math. Comp. 90 (2021), 2671-2703
MSC (2020): Primary 65N06, 35J70, 90C49
Published electronically: July 15, 2021
MathSciNet review: 4305365
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Abstract: We design adaptive finite differences discretizations, which are degenerate elliptic and second order consistent, of linear and quasi-linear partial differential operators featuring both a first order term and an anisotropic second order term. Our approach requires the domain to be discretized on a Cartesian grid, and takes advantage of techniques from the field of low-dimensional lattice geometry. We prove that the stencil of our numerical scheme is optimally compact, in dimension two, and that our approach is quasi-optimal in terms of the compatibility condition required of the first and second order operators, in dimensions two and three. Numerical experiments illustrate the efficiency of our method in several contexts.

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Additional Information

J. Frédéric Bonnans
Affiliation: Inria-Saclay and CMAP, Ecole Polytechnique, Palaiseau, France

Guillaume Bonnet
Affiliation: LMO, Université Paris-Saclay, Orsay, France; and Inria-Saclay and CMAP, Ecole Polytechnique, Palaiseau, France

Jean-Marie Mirebeau
Affiliation: University Paris-Saclay, ENS Paris-Saclay, CNRS, Centre Borelli, F-91190 Gif-sur-Yvette, France
MR Author ID: 908991
ORCID: 0000-0002-7479-0485

Keywords: Degenerate ellipticity, monotone finite difference schemes, Voronoi’s first reduction
Received by editor(s): December 20, 2020
Received by editor(s) in revised form: March 8, 2021
Published electronically: July 15, 2021
Additional Notes: The first author was supported by the Chair Finance & Sustainable Development and of the FiME Lab (Institut Europlace de Finance). This work was partly supported by ANR research grant MAGA, ANR-16-CE40-0014.
Article copyright: © Copyright 2021 American Mathematical Society