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Face-to-face partition of 3D space with identical well-centered tetrahedra. (English) Zbl 1363.65209
Summary: The motivation for this paper comes from physical problems defined on bounded smooth domains \(\Omega \) in 3D. Numerical schemes for these problems are usually defined on some polyhedral domains \(\Omega_h\) and if there is some additional compactness result available, then the method may converge even if \(\Omega_h \to \Omega \) 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.

65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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