On the sparsity patterns of hierarchical finite element matrices
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The performance of preconditioned conjugate gradient methods for the solution of a linear system of equations Hx=b depends strongly on the quality of the preconditioner. In many applications the system of equations to be solved originates from a partial differential equation discretized by the choice of an initial mesh geometry, a meshrefinement technique and a type of finite element basisfunctions. In general the resulting matrix is a sparse matrix which sparsity pattern only depends on the discretization choices. For the construction of a preconditioner only the matrix entries are needed, but investigations so far clearly have shown that taking into account additionally the discretization choices via the sparsity pattern leads to more effective preconditioning techniques, of which many exist.
Relatively simple techniques like ILU and SSOR take the matrix entries into account but disregard the sparsity pattern. Somewhat more effective techniques like Block-Incomplete ones make use of the sparsity patterns regularity resulting from a regular mesh geometry. In addition multi-grid techniques take into account that a fine mesh geometry is obtained by the refinement of a coarse mesh. Intermediate ‘levels’ of refinement are distinguished and used explicitly. More flexible are algebraic multilevel preconditioners which assign a ‘level’ to each individual degree of freedom.
As the sparsity pattern is of importance for the construction of good preconditioners it is analysed for the hierarchical matrix H resulting from a given discretization. The hierarchy is induced by the mesh refinement method applied. It is shown that the sparsity pattern is irregular but well structured in general and a simple refinement method is presented which enables a compact storage and quick retrieval of the matrix entries in the computers memory. An upperbound for the C.-B.-S. scalar for this method is determined to demonstrate that it is well suited for multilevel preconditioning and it is shown to have satisfying angle bounds. Further it turns out that the hierarchical matrix may be partially constructed in parallel, is block structured and shows block decay rates.
KeywordsSparsity structure Adaptive mesh refinement Hierarchical finite elements Error indication Parallel computing AMS(MOS) subject classifications 65N30 65N50 65W05 65F10 65F10 65F50
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- Axelsson O, An algebraic framework for multilevel methods, internal report 8820 (October 1988), Department of Mathematics, University of Nijmegen, The NetherlandsGoogle Scholar
- Axelsson O. and Vassilevski P.S., Algebraic multilevel preconditiong methods II, Siam J. Numer.Anal. 1990, to appearGoogle Scholar
- Bank R.E., and Dupont T.F., Analysis of a two-level scheme for solving finite element equations, Report CNA-159, Center for Numerical Analysis, University of Texas, Austin, 1980Google Scholar
- Bank R.E., Dupont T.F. and Yserentant H., The hierachical basis multigrid method, Preprint SC 87-1 (1987), Konrad-Zuse-Zentrum für Informationstechnik, BerlinGoogle Scholar
- Bank R.E. and Sherman A.H., The use of adaptive grid refinement for badly behaved elliptic partial differential equations, in Advances in Computer Methods for Partial Differential Equations II (Vichnevetsky R. and Stepleman R.S. eds.), IMACS, 1979, 33–39Google Scholar
- Becker E.B., Finite Elements, Prentice Hall, Englewood Cliffs, New Jersey, 1981Google Scholar
- Carey G.F., Sharma M. and Wang K.C., A class of data structures for 2-D and 3-D adaptive mesh refinement, in press, International Journal for Numerical Methods in EngineeringGoogle Scholar
- Greenbaum A., Congming Li and Han Zheng Chao, Comparison of linear system solvers applied to diffusion-type finite element equations, internal report Ultracomputer Note #126, New York University, September 1987Google Scholar
- Maitre J.F. and Musy F., The contraction number of a class of two-level methods; an exact evalutation for some finite element subspaces and model problems, in Multigrid Methods (Hackbush W. and Trottenberg U. eds.), Proceedings Köln-Porz 1981, LNM 960, Springer Verlag, 1982, 535–544Google Scholar
- Maubach J.M., On the sparsity pattern, block decay rates and preconditioning of multilevel hierarchical finite element matrices, in preparation, intern. report, University of Nijmegen 1989Google Scholar
- Mitchell W.F., A comparison of adaptive refinement techniques for elliptic problems, internal report no. UIUCDCS-R-1375, Department of computer science, University of Illinois at Urbana-Champaign, Urbana, Illinois(1987), to appear in ACM Trans. Math. Soft.Google Scholar
- Mitchell W.F., Unified multilevel adaptive finite element methods for elliptic problems, internal report no. UIUCDCS-R-88-1436, Department of computer science, University of Illinois at Urbana-Champaign, Urbana, Illinois(1988)Google Scholar
- Sewell E.G., Automatic generation of triangulations for piecewise polynomial approximation, Ph.D. Thesis, Purdue University, West Lafayette, IN, 1972Google Scholar
- Sewell E.G., A finite element program with automatic user-controlled mesh grading, in Advances in Computer Methods for Partial Differential Equations III (Vichnevetsky R. and Stepleman R.S. eds.), IMACS, 1979, 8–10Google Scholar
- Zienkiewicz O., The Finite Element Method in Engineering Science, 3rd edition, Mc Graw-Hill, New York, 1977Google Scholar