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Accueil du site > Résumés des séminaires > Labo > Minimal stencils for structure preserving discretizations of Anisotropic PDEs

Minimal stencils for structure preserving discretizations of Anisotropic PDEs

I will present monotone discretizations of anisotropic diffusion, and causal discretizations of anisotropic eikonal equations, on two and three dimensional grids. The proposed schemes rely on sparse and anisotropic stencils, adapted to the underlying diffusion tensor or Riemannian metric, and built using Lattice Basis Reduction - a tool from discrete geometry commonly used in integer programming, cryptography or number theory, but only recently introduced in the field of numerical PDEs.

The stencils of these numerical schemes not only have bounded cardinality, but are the smallest possible geometrically speaking, in sense that the convex envelope of the stencil points is minimal (for inclusion) among all monotone (resp. causal) grid discretizations of anisotropic diffusion (resp. eikonal) equation. We also discuss the maximum and average radii of these stencils, which have wildly different behaviors, and present an equivalence result, for anisotropic diffusion, with an adaptive finite element discretization on an anisotropic Delaunay triangulation.


CMAP UMR 7641 École Polytechnique CNRS, Route de Saclay, 91128 Palaiseau Cedex France, Tél: +33 1 69 33 46 23 Fax: +33 1 69 33 46 46