Direct Solution of the Discretized Poisson-Neumann Problem on a Domain Composed of Rectangles

Abstract

Finite-difference approximations for the two-dimensional Poisson-Neumann problem (l/λ) div λ grad p = q, λ>0 on R (λ_n x grad p = g_R on ∂R result in a large system of linear equations Lp = q. The n x n matrix L is singular with rank(L) = n - 1. A solution exists if q satisfies the consistency condition v^Tq = 0, where L^Tv = 0, v ≠ 0. A direct-solution scheme is described for the case where R can be decomposed into a set of rectangular domains each having a possibly different but constant material coefficient λ. We assume that this problem has to be solved repeatedly for many vectors q. The method is efficient in that the number of operations is of order n log n. This efficiency is gained by using fast elliptic solvers for each single rectangle and a proper variant of the socalled influence- or capacitance-matrix technique. Here, on each rectangle a separate Poisson-Neumann problem has to be solved. The essential point is the manner in which the consistency condition is enforced for each rectangle. The new method has been successfully applied in a code FLUX to analyze fluid-structure interactions.