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Keywords:
rigid mesh; well-centered mesh; approximative domain; single element mesh; Sommerville tetrahedron
rigid mesh; well-centered mesh; approximative domain; single element mesh; Sommerville tetrahedron
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. \endgraf Numerical schemes for which quantities are defined on dual partitions usually require some additional quality. One of the used approaches is the concept of \emph {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.
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