Monotone Discretization of Anisotropic Differential Operators Using Voronoi’s First Reduction

Abstract

We consider monotone discretization schemes, using adaptive finite differences on Cartesian grids, of partial differential operators depending on a strongly anisotropic symmetric positive definite matrix. For concreteness, we focus on a linear anisotropic elliptic equation, but our approach extends to divergence form or non-divergence form diffusion, and to a variety of first and second order Hamilton-Jacobi-Bellman PDEs. The design of our discretization stencils relies on a matrix decomposition technique coming from the field of lattice geometry, and related to Voronoi's reduction of positive quadratic forms. We show that it is efficiently computable numerically, in dimension up to four, and yields sparse and compact stencils. However, some of the properties of this decomposition, related with the regularity and the local connectivity of the numerical scheme stencils, are far from optimal. We thus present fixes and variants of the decomposition that address these defects, leading to stability and convergence results for the numerical schemes.