SOLVING DIFFUSION-EQUATIONS WITH ROUGH COEFFICIENTS IN ROUGH GRIDS

A finite-difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and nonisotropic media is construct ed for logically rectangular grids. The performance of this algorithm is comparable to other algorithms for problems with smooth coefficient s and regular grids, and it is superior for problems with rough coeffi cients and/or skewed grids. The algorithm is derived using the support -operators method, which constructs discrete analogs of the divergence and flux operator that satisfy discrete analogs of the important inte gral identities relating the continuum operators. This paper gives the first application of this method to the solution of diffusion problem s in heterogeneous an nonisotropic media. The support-operators method forces the discrete analog of the flux operator to be the negative ad joint of the discrete divergence in an inner product weighted by the c onductivity, as in the differential case. Once this is accomplished, m any other important properties follow; for example, the scheme is cons ervative and the discrete analog of the variable material Laplacian is symmetric and negative definite. In addition, on any grid, the discre te divergence is zero on constant vectors and the discrete flux operat or is exact for linear functions in case when K is piecewise constant. Moreover, the discrete gradient's null space is the constant function s, just as in the continuum. Because the algorithm is flux based, it h as twice as many unknowns as more standard algorithms. However, the ma trices that need to be inverted are symmetric and positive definite, s o the most powerful linear solvers can be applied. Also, the scheme is second-order accurate so, all things considered, it is efficient. For rectangular grids, the discrete operators reduce to well-known discre te operators and the treatment of discontinuous conductivity coefficie nts in the case of isotropic media is equivalent to the well-known har monic-averaging procedure. Comparison with standard schemes is present ed. Numerical examples validate advantage of new method. (C) 1996 Acad emic Press, Inc.