On the Radius-Edge Condition in the Control Volume Method

In this paper we show that the control volume algorithm for the solution of Poisson's equation in three dimensions will converge even if the mesh contains a class of very flat tetrahedra (slivers). These tetrahedra are characterized by the fact that they have modest ratios of diameter to shortest edge, but large circumscribing to inscribed sphere radius ratios, and therefore may have poor interpolation properties. Elimination of slivers is a notoriously difficult problem for automatic mesh generation algorithms. We also show that a discrete Poincaré inequality will continue to hold in the presence of slivers.