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Second order monotone finite differences discretization of linear anisotropic differential operators
Author(s) -
J. Frédéric Bonnans,
Guillaume Bonnet,
Jean-Marie Mirebeau
Publication year - 2021
Publication title -
mathematics of computation
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.95
H-Index - 103
eISSN - 1088-6842
pISSN - 0025-5718
DOI - 10.1090/mcom/3671
Subject(s) - stencil , discretization , mathematics , differential operator , monotone polygon , finite element method , lattice (music) , cartesian coordinate system , elliptic operator , mathematical analysis , regular grid , grid , geometry , physics , computational science , acoustics , thermodynamics
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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