  We introduce a new discretization scheme for Anisotropic Diffusion, AD-LBR,
on two and three dimensional cartesian grids. The main features of this scheme
is that it is non-negative, and has a stencil cardinality bounded by 6 in 2D,
by 14 in 3D, despite allowing diffusion tensors of arbitrary anisotropy. Our
scheme also has good spectral properties, which permits larger time steps and
avoids e.g. chessboard artifacts.
  AD-LBR relies on Lattice Basis Reduction, a tool from discrete mathematics
which has recently shown its relevance for the discretization on grids of
strongly anisotropic Partial Differential Equations. We prove that AD-LBR is in
2D asymptotically equivalent to a finite element discretization on an
anisotropic Delaunay triangulation, a procedure more involved and
computationally expensive. Our scheme thus benefits from the theoretical
guarantees of this procedure, for a fraction of its cost. Numerical experiments
in 2D and 3D illustrate our results.
