Abstract
The steady state diffusion equation may be written: (1)Va-Vu+aAu=f for the case in which the diffusivity a varies spatially. In contrast to the "forward" problem, a second order elliptic partial differential equation for u, the "inverse" problem is a first order hyperbolic problem, in which one attempts to determine a from an observed solution u produced by a known f.A distinguishing feature of this hyperbolic problem is that its coefficients 7u and Au must be approximated numerically, since normally oRly u would be directly available from experimental data. In [2], we developed a numerical discretization for the inverse problem using a central difference approximation to Au and upwind approximations to Va and Vu, modified so as to produce an explicit scheme in the direction of increasing grid values of u. Implementation of the scheme begins with a sorting of these values, and the discrete solution is then obtained in a pattern determined by the characteristics of the hyperbolic problem for a. It was also shown that the scheme achieves its optimal O(h) convergence Tate in the case where a and u are sufficiently smooth, provided Au is positive throughout the domain. In this paper, we analyze the performance of the same scheme under the assumption that 17ul is bounded away from zero. This assumption guarantees that the characteristics of the inverse problem are everywhere well-defined and nonintersecting, so that the inverse problem is well-set, subject to specification of appropriate boundary values for a. Thus one would expect that the numerical scheme, a consistent discretization that violates no domain of dependence criteria, should perform optimally. However, we find it necessary to impose a condition restricting the number of sign variations in the components If Vu in order to insure that this is the case, the additional condition precludes the possibility of an infinitely meandering "discrete characteristic" as the grid size approaches zero. In our analysis, we use a maximum principle (2) approach similar to that suggested by Godunov and Ryabenki [1] for constant coefficient hyperbolic problems. For the case in which a is known only to be positive, we have no recipe for choosing a forcing function and boundary conditions so as to produce a solution to the forward problem for which Au>O or IVul>O over the entire domain. However, if f is positive, which is achievable in principle, then one may divide (3) the domain into sub domains in which one condition or the other is satisfied, using the positivity of a. Thus we anticipate that by a local patching together of the results in this paper and its predecessor, we shall obtain a continuous dependence result for the inverse problem and a proof of convergence for the numerical scheme in the case where f>O. We defer this to a subsequent paper.