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Unformatted text preview: Conforming discretizations on tetrahedrons, pyramids, prisms and hexahedrons Christian Wieners Institut f¨ur Computeranwendungen III, Universit¨at Stuttgart Pfaffenwaldring 27, 70569 Stuttgart, Germany Abstract We describe conforming P and P discretizations on tetrahedrons, pyramids, prisms and hexahedrons and we prove the stability of the corresponding TaylorHood elements. They can be used for flexible approximations of complicated geometries on locally refined grids. We give examples for the discretization of a diffusion equation and of a problem in linear elasticity. The discretized problems are solved with an adaptive multigrid method. AMS Subject Classification: 65N30 Key words: finite elements, multigrid methods Introduction In many three dimensional applications hexahedral grids are very popular. They provide better approximations than a corresponding tetrahedral grid with the same number of cor ners. Nevertheless, there are disadvantages for the hexahedrons: it is difficult to represent arbitrary geometries and one obtains hanging nodes in a local refinement procedure. An elegant alternative is the flexible use of different types of elements. In particular, mesh generation with hexahedral grids is a very difficult task. In order to avoid elements with bad angles it is required to use other element types as well. In general, consistent grids combining hexahedrons and tetrahedrons can only be obtained if pyramids are included. (A grid is consistent , if the intersection of two elements is empty, a common node, a common edge or a common side.) Finally, efficient approximations require a locally refined grid which can be adapted in every step for time depending problems. Algorithms for refinement and derefinement of grids of this type are introduced by LANG [5]. On consistent grids conforming approximations are desirable. Therefore, we give a list of shape functions for two basic discretizations on tetrahedrons, pyramids, prisms and he xahedrons. For hexahedrons and tetrahedrons we use the standard shape functions, the shape functions of pyramids and prisms are constructed in order to obtain globally con tinuous approximations. For pyramids, the construction needs a splitting into two tetrahe drons. In addition, we prove that there are no polynomial shape function for pyramids. For problems in fluid dynamics, TaylorHood elements are popular in two dimensions, but they are not very often used in three dimensions. The stability is proved for tetrahedrons only (cf. VERF ¨UHRT [6, 3]). The extension of the stability proof to pyramids, prisms and hexahedrons is not straight forward and requires the explicit construction of the shape functions. 2 C. Wieners We do not restrict to the standard multilinear discretizations only because they have poor approximation properties. Here, we demonstrate this drawback and on a problem in linear elasticity for a cube: due to locking effects the slow pointwise convergence for the dis placement leads to an enormous memory requirement to obtain a reasonable accuracy. Weplacement leads to an enormous memory requirement to obtain a reasonable accuracy....
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This note was uploaded on 04/02/2012 for the course MA 125 taught by Professor Dr.david during the Spring '10 term at Ecole Polytechnique Fédérale de Lausanne.
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