Domain Contractions around Three-Dimensional Anomalies in Spherical Finite-Difference Computations of Poisson’s Equation

It is computationally advantageous to solve a finite-difference approximation to Poisson’s equation ∇ · (σ ∇ P) + H = 0 in an infinite domain, wherein σ varies with all three spherical coordinates inside a sphere SJ but only with the radial and latitudinal coordinates outside SJ, and H vanishes outside SJ by representing the medium outside SJ, by a driving-point conductance matrix. This is equivalent to replacing the spherical grid outside SJ by a set of terminating conductances connected to the nodes on SJ. Thus, the large system matrix generated by the customary truncation of the infinite medium far beyond SJ is replaced by the much smaller matrix corresponding to the nodes on and within SJ. This technique is especially advantageous when the problem is to be solved many times for various hypothesized variations of σ within SJ, because the terminating conductances depend only upon the values of σoutside SJ. An operational calculus for circulant matrices expedites the computation of the terminating conductances. The method is illustrated by computing the flow of petroleum through the completion zone of an oil well. A proof of the convergence of the method is given for the case where as much current is extracted from the medium as is injected into it.