Abstract
The partial differential equation Div (a grad u) = f is the governing equation for isotropic steady state diffusion processes. Among its applications are heat conduction, the dispersion of fluids of varying concentration, and fluid flow through porous media. Typically one wishes to determine a "solution" u for a prescribed f, subject to certain boundary conditions. The "diffusivity" is a an intrinsic property of the medium in which the diffusion process occurs. In many cases, it varies spatially in a manner which one can hope to determine only by measuring the solution produced by a known f and then attempting to solve the inverse problem, which amounts to a first order hyperbolic partial differential equation for a. Considerable attention has been paid to this inverse problem, especially within the context of underground flow of fluids through porous media. This is exemplified by the work of Frind and Pinder [2], who describe a Galerkin procedure for the inverse problem and also impart a sense of perspective on the problem from a hydrogeologic viewpoint. From a more theoretical vantage point, Chavent [1] has established well-posed ness of the inverse problem in the Lw sense in the case where the Laplacian of u is bounded away from zero. Our basic objective is to provide a numerical scheme for the inverse problem whose stability and convergence can be guaranteed under appropriate conditions. We independently establish Chavent's continuous dependence result via a quite straightforward maximum principle approach. A discrete version of the maximum principle is then used to prove O(h) convergence for a modified Upwind-downwind differencing scheme depending on the sign of the Laplacian of U. The scheme is explicit in the direction of increasing grid values of U. Its implementation is thus preceded by a sorting of these values, whereupon the discrete solution is developed in a pattern determined by the characteristics of the continuous problem for a. The scheme requires initial data for a along the "inflow" portion of the boundary, and for problems in which there is no inflow, it generates initial data internally at points where Vu=o. Several illustrative examples are included.