Abstract

We present new geometrical and numerical analysis structure theorems for the Delaunay diagram of point sets in IR d for a fixed d where the point sets arise naturally in numerical methods. In particular, we show that if the largest ratio of the circum-radius to the length of smallest edge over all simplexes in the Delaunay diagram of P , DT (P ), is bounded, (called the bounded radius-edge ratio property), then DT (P ) is a subgraph of a density graph, the Delaunay spheres form a k-ply system for a constant k, and that we get optimal rates of convergence for approximate solutions of Poisson's equation constructed using control volume techniques. The density graph result implies that DT (P ) has a partition of cost O(n 1\Gamma1=d ) that can be efficiently found by the geometric separator algorithm of Miller, Teng, Thurston, and Vavasis and therefore the numerical linear system defined on DT (P) using the finite-volume method can be solved efficiently on a parallel machine (either by...

Citations