Discrete Sobolev-Poincaré Inequalities for Voronoi Finite Volume Approximations

We prove a discrete Sobolev-Poincaré inequality for functions with arbitrary boundary values on Voronoi finite volume meshes. We use Sobolev’s integral representation and estimate weakly singular integrals in the context of finite volumes. We establish the result for star shaped polyhedral domains and generalize it to the finite union of overlapping star shaped domains. In the appendix we prove a discrete Poincaré inequality for space dimensions greater or equal to two.