The Numerical Solution of Diffusion Problems in Strongly Heterogeneous Non-isotropic Materials

A new second-order finite-difference algorithm for the numerical where nW is the vector of unit outward normal to the boundsolution of diffusion problems in strongly heterogeneous and nonary ­V, and a and c are functions given on ­V. The algoisotropic media is constructed. On problems with rough coefficients or highly nonuniform grids, the new algorithm is superior to all rithm is constructed using a nontrivial generalization of other algorithms we have compared it with. For problems with the support-operators method for solving problems where smooth coefficients on smooth grids, the method is comparable the material properties tensor (or matrix) K may be disconwith other second-order methods. The new algorithm is formulated tinuous and non-diagonal and, moreover, the computafor logically rectangular grids and is derived using the supporttional grid may not be smooth. operators method. A key idea in deriving the method was to replace the usual inner product of vector functions by an inner product The support-operators method constructs discrete anaweighted by the inverse of the material properties tensor and to logs of invariant differential operators div and grad, which use the flux operator, defined as the material properties tensor times satisfy discrete analogs of the integral identities responsible the gradient, rather than the gradient, as one of the basic first-order for the conservative properties of the continuum model. operators in the support-operators method. The discrete analog of The method was initially developed in [1] by Samarskii, the flux operator must also be the negative adjoint of the discrete divergence, in an inner product that is a discrete analog of the Tishkin, Favorskii, and Shashkov and is fully described continuum inner product. The resulting method is conservative and in [2]. the discrete analog of the variable coefficient Laplacian is symmetric This paper is the third of a series on the support-operaand negative definite on nonuniform grids. In addition, on any grid, tors method. In the first paper [3], the support-operators the discrete divergence is zero on constant vectors, the null space method was combined with the mapping method to profor the gradient is the constant functions, and, when the material properties are piecewise constant, the discrete flux operator is exact duce an algorithm for equations with general boundary for piecewise linear functions. We compare the methods on some conditions. The resulting method was shown to be accurate of the most difficult examples to be found in the literature. Q 1997 when both K is smooth and the problem is solved on a Academic Press smooth grid. In the second paper [4], the support-operators method was extended to define a new cell-centered finitedifference algorithm for solving time-dependent diffusion