The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials

A new second-order finite-difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and nonisotropic media is constructed. On problems with rough coefficients or highly nonuniform grids, the new algorithm is superior to all other algorithms we have compared it with. For problems with smooth coefficients on smooth grids, the method is comparable with other second-order methods. The new algorithm is formulated for logically rectangular grids and is derived using the support-operators method. A key idea in deriving the method was to replace the usual inner product of vector functions by an inner product weighted by the inverse of the material properties tensor and to use the flux operator, defined as the material properties tensor times the gradient, rather than the gradient, as one of the basic first-order operators in the support-operators method. The discrete analog of the flux operator must also be the negative adjoint of the discrete divergence, in an inner product that is a discrete analog of the continuum inner product. The resulting method is conservative and the discrete analog of the variable coefficient Laplacian is symmetric and negative definite on nonuniform grids. In addition, on any grid, the discrete divergence is zero on constant vectors, the null space for the gradient is the constant functions, and, when the material properties are piecewise constant, the discrete flux operator is exact for piecewise linear functions. We compare the methods on some of the most difficult examples to be found in the literature. (C) 1997 Academic Press.