Refinement strategies related to cubic tetrahedral meshes

The paper deals with refinement techniques suitable for application of finite element multigrid methods with cubic trial functions. The 27-refinement strategy by Edelsbrunner and Grayson has not been studied from computational point of view up to now. This refinement strategy is said to be dark red refinement strategy (DRRS) by analogy with the red refinement strategy in the quadratic case. A detail analysis of DRRS is an aim of this paper. The dependence of the DRRS on the numbering of vertex nodes requires an algorithm development. Freudenthal partition of the cube has been widely used by researchers for obtaining hierarchical tetrahedral triangulations. In the cubic case, the refinement of the first Sommerville tetrahedron is of considerable practical importance. Since the DRRS generates one class of similarity only with a particular numbering of nodes, we have developed a detailed algorithm for proper implementation of this numbering. The situation is much more complicated when an arbitrary tetrahedron is refined. A large volume of computations are necessary in order to be established a correct numbering of the vertex nodes. If the best numbering of nodes is not chosen for all elements in all levels, the measure of degeneracy tends to infinity when the number of levels grows up unlimited. To avoid these difficulties a new canonical refinement strategy (CRS) is obtained. The CRS is superior compared to the 27-partition technique with respect to the degeneracy measure. It generates only regular tetrahedra and canonical simplices in all levels and for all mixed domains. These two tetrahedra are the most convenient from a computational point of view. Moreover, for all mixed domains the CRS essentially reduces the number of congruence classes. The new refinement strategy does not need an algorithm for correct numbering of the vertex nodes. The DRRS generates a very complicated refinement tree and a higher measure of degeneracy in combination with the face centered partition of the cube. On the contrary the Z refinement strategy obtained by the authors is superior than the DRRS in regard to the number of congruence classes and the measure of degeneracy. Some results on boundary value problems in changing domains are discussed. The advantages of the CRS are demonstrated by solving an anisotropic diffusion problem in a shrinking domain. The behavior of discretization and truncation errors in approximate finite element solutions is illustrated by varying sequences of finite element triangulations. (C) 2018 IMACS. Published by Elsevier B.V. All rights reserved.