We consider a quantity that measures the roundness of a bounded, convex @math -polytope in @math . We majorise this quantity in terms of the smallest singular value of the matrix of outer unit normals to the facets of the polytope.
Similar quantities appear also in the literature on finite volume methods. For example, in @cite_3 , a finite volume method based on Voronoi tessellations of the sphere is constructed, and a regularity norm' is defined for the Voronoi tessellation by minimising the ratio @math over all neighbours of each Voronoi cell @math generated by a point @math on the sphere, and over all Voronoi cells @math in the tessellation. Note that in both @math and @math above are computed with respect to the geodesic metric on the sphere. For any given Voronoi cell @math with generator @math , the smallest value of @math over all generators @math of adjacent Voronoi cells @math resembles the degeneracy ratio of @math defined in . This is because if @math is a Voronoi cell with generator @math , and if @math is small, then @math will be small, and vice versa.
@cite_3
2161434505
We first develop and analyze a finite volume scheme for the discretization of partial differential equations (PDEs) on the sphere; the scheme uses Voronoi tessellations of the sphere. For a model convection–diffusion problem, the finite volume scheme is shown to produce first-order accurate approximations with respect to a mesh-dependent discrete firstderivative norm. Then, we introduce the notion of constrained centroidal Voronoi tessellations (CCVTs) of the sphere; these are special Voronoi tessellation of the sphere for which the generators of the Voronoi cells are also the constrained centers of mass, with respect to a prescribed density function, of the cells. After discussing an algorithm for determining CCVT meshes on the sphere, we discuss and illustrate several desirable properties possessed by these meshes. In particular, it is shown that CCVT meshes define very high-quality uniform and non-uniform meshes on the sphere. Finally, we discuss, through some computational experiments, the performance of the CCVT meshes used in conjunction with the finite volume scheme for the solution of simple model PDEs on the sphere. The experiments show, for example, that the CCVT based finite volume approximations are second-order accurate if errors are measured in