Second order monotone finite differences discretization of linear anisotropic differential operators

Frédéric Bonnans (1) , Guillaume Bonnet (2, 1) , Jean-Marie Mirebeau (3, 4)

1 COMMANDS - Controle, Optimisation, modèles, Méthodes et Applications pour les Systèmes Dynamiques non linéaires
2 LMO - Laboratoire de Mathématiques d'Orsay
3 CB - CB - Centre Borelli - UMR 9010
4 CNRS - Centre National de la Recherche Scientifique

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 dimension two and three. Numerical experiments illustrate the efficiency of our method in several contexts.

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Frédéric Bonnans, Guillaume Bonnet, Jean-Marie Mirebeau. Second order monotone finite differences discretization of linear anisotropic differential operators. Mathematics of Computation, 2021, Mathematics of Computation, 90, pp.2671-2703. ⟨10.1090/mcom/3671⟩. ⟨hal-03084046v2⟩