  Nonnegative directional splittings of anisotropic diffusion operators in the
divergence form are investigated. Conditions are established for nonnegative
directional splittings to hold in a neighborhood of an arbitrary interior
point. The result is used to construct monotone finite difference schemes for
the boundary value problem of anisotropic diffusion operators. It is shown that
such a monotone scheme can be constructed if the underlying diffusion matrix is
continuous on the closure of the physical domain and symmetric and uniformly
positive definite on the domain, the mesh spacing is sufficiently small, and
the size of finite difference stencil is sufficiently large. An upper bound for
the stencil size is obtained, which is determined completely by the diffusion
matrix. Loosely speaking, the more anisotropic the diffusion matrix is, the
larger stencil is required. An exception is the situation with a strictly
diagonally dominant diffusion matrix where a three-by-three stencil is
sufficient for the construction of a monotone finite difference scheme.
Numerical examples are presented to illustrate the theoretical findings.
