×

zbMATH — the first resource for mathematics

Tetrahedral grid refinement. (English) Zbl 0839.65135
Adaptive techniques in the finite element methods optimize the number of unknowns by fitting the corresponding discretization. In three-dimensional problems, the underlying discretization mesh is given by a triangulation with the help of tetrahedral grids. Thus, the author considers an adaptive method to generate sequences of triangulations which are consistent (roughly spoken, this is the prevention of so-called “hanging nodes”) and stable (the interior angles of all elements are uniformly bounded away from zero).
To make compatible adaptivity, consistency, and stability the refinement algorithm is constructed in three steps. First, a basic strategy is defined for the subdivision of a tetrahedron into eight subtetrahedra of equal volume (a regular strategy). Second, irregular refinement rules are chosen for elements not refined regularly but sharing a refined edge or side of another element. Finally, a global refinement algorithm describes how the local rules can be combined. A routine and subroutines of the whole algorithm are formulated independently of any programming language – a useful tool to understand the construction in all details.

MSC:
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Software:
PLTMG
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Baensch, E.: Local mesh refinement in 2 and 3, dimensions. Impact Comput. Sci. Eng.3, 181–191 (1991). · Zbl 0744.65074 · doi:10.1016/0899-8248(91)90006-G
[2] Bank, R.E.: PLTMG: A software package for solving elliptic partial differential equations, Users’ Guide 6.0. Philadelphia: SIAM, 1990. · Zbl 0717.68001
[3] Bank, R.E., Sherman, A.H., Weiser, A.: Refinement algorithms and data structures for regular local mesh refinement, In: Scientific computing (Stepleman, R., ed.) pp. 3–17 Amsterdam: IMACS/North-Holland, 1983.
[4] Bastian, P.: Parallele adaptive Mehrgitterverfahren. PhD thesis, Univ. Heidelberg, 1994. · Zbl 0818.65119
[5] Bey, J.: Analyse und Simulation eines Konjugierte-Gradienten-Verfahrens mit einem Multilevel Präkonditionierer zur Lösung dreidimensionaler elliptischer Randwertprobleme für massiv parallele Rechner. Master’s thesis, Institut für Geometrie und Praktische Mathematik, RWTH Aachen, 1991.
[6] Bey, J.: A robust multigrid method for 3d convection-diffusion equations. PhD thesis, Univ. Tübingen (in preparation).
[7] Bey, J.: AGM3D Manual. Tech. Rep., Univ. Tübingen, 1994.
[8] Bornemann, F. A., Erdmann, B., Kornhuber, R.: Adaptive multilevel methods in three space dimensions. Int. J. Numer. Meth. Eng.36, 3187–3203 (1993). · Zbl 0780.73073 · doi:10.1002/nme.1620361808
[9] Bramble, J.H., Pasciak, J.E., Wang, J., Xu, J.: Convergence estimates for multigrid algorithms without regularity assumptions. Math. Comp.57, 23–45 (1991). · Zbl 0727.65101 · doi:10.1090/S0025-5718-1991-1079008-4
[10] Ciarlet, P. G.: The finite element method for elliptic problems. Amsterdam: North-Holland, (1978). · Zbl 0383.65058
[11] Freudenthal, H.: Simplizialzerlegungen von beschränkter Flachheit. Ann. Math.43, 580–582 (1992). · Zbl 0060.40701 · doi:10.2307/1968813
[12] Hackbusch, W.: Multigrid methods and applications. Berlin Heidelberg New York Tokyo: Springer, 1985. · Zbl 0595.65106
[13] Kuhn, H.W.: Some combinatorial lemmas in topology. IBM J. Res. Dev.45, 518–524 (1960). · Zbl 0109.15603 · doi:10.1147/rd.45.0518
[14] Leinen, P.: Data structures and concepts for adaptive finite element methods. Computing55, 325–354 (1995). · Zbl 0837.65133 · doi:10.1007/BF02238486
[15] Maubach, J.M.L.: Local bisection refinement for N-simplicial grids generated by reflection. SIAM J. Sci. Comput.16, 210–227 (1995). · Zbl 0816.65090 · doi:10.1137/0916014
[16] Mitchell, W.F.: Adaptive refinement for arbitrary finite-element spaces with hierarchical basis. J. Comput. Appl. Math.36, 65–78 (1991). · Zbl 0733.65066 · doi:10.1016/0377-0427(91)90226-A
[17] Rivara, M.C.: Design and data structure of a fully adaptive multigrid finite element software. ACM Trans. Math. Software10, 242–264 (1989). · Zbl 0578.65112 · doi:10.1145/1271.1274
[18] Yserentant, H.: Old and new convergence proofs of multigrid methods. Acta Numer. 285–326 (1993). · Zbl 0788.65108
[19] Zhang, S.: Multin-level iterative techniques. PhD thesis, Research Report no. 88020, Dept. of Math., Pennstate Univ., 1988.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.