Solving Diffusion Equations with Rough Coefficients in Rough Grids

Abstract

A finite-difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and nonisotropic media is constructed for logically rectangular grids. The performance of this algorithm is comparable to other algorithms for problems with smooth coefficients and regular grids, and it is superior for problems with rough coefficients and/or skewed grids. The algorithm is derived using thesupport-operators method,which constructs discrete analogs of the divergence and flux operator that satisfy discrete analogs of the important integral identities relating the continuum operators. This paper gives the first application of this method to the solution of diffusion problems in heterogeneous an nonisotropic media.The support-operators method forces the discrete analog of the flux operator to be the negative adjoint of the discrete divergence in an inner product weighted by the conductivity, as in the differential case. Once this is accomplished, many other important properties follow; for example, the scheme is conservative and the discrete analog of the variable material Laplacian is symmetric and negative definite. In addition, on any grid, the discrete divergence is zero on constant vectors and the discrete flux operator is exact for linear functions in case whenKis piecewise constant. Moreover, the discrete gradient's null space is the constant functions, just as in the continuum. Because the algorithm is flux based, it has twice as many unknowns as more standard algorithms. However, the matrices that need to be inverted are symmetric and positive definite, so 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 discrete operators and the treatment of discontinuous conductivity coefficients in the case of isotropic media is equivalent to the well-known harmonic-averaging procedure. Comparison with standard schemes is presented. Numerical examples validate advantage of new method.