Face-to-face partition of 3D space with identical well-centered tetrahedra

Abstract

The motivation for this paper comes from physical problems defined on bounded smooth domains Ω in 3D. Numerical schemes for these problems are usually defined on some polyhedral domains Ω _h and if there is some additional compactness result available, then the method may converge even if Ω _h → Ω only in the sense of compacts. Hence, we use the idea of meshing the whole space and defining the approximative domains as a subset of this partition.

Numerical schemes for which quantities are defined on dual partitions usually require some additional quality. One of the used approaches is the concept of well-centeredness, in which the center of the circumsphere of any element lies inside that element. We show that the one-parameter family of Sommerville tetrahedral elements, whose copies and mirror images tile 3D, build a well-centered face-to-face mesh. Then, a shape-optimal value of the parameter is computed. For this value of the parameter, Sommerville tetrahedron is invariant with respect to reflection, i.e., 3D space is tiled by copies of a single tetrahedron.