On the sparsity patterns of hierarchical finite element matrices

  • J. Maubach
Submitted Papers
Part of the Lecture Notes in Mathematics book series (LNM, volume 1457)


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.


Sparsity structure Adaptive mesh refinement Hierarchical finite elements Error indication Parallel computing AMS(MOS) subject classifications 65N30 65N50 65W05 65F10 65F10 65F50 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Axelsson O, An algebraic framework for multilevel methods, internal report 8820 (October 1988), Department of Mathematics, University of Nijmegen, The NetherlandsGoogle Scholar
  2. [2]
    Axelsson O. and Barker V.A., Finite Element Solution of Boundary Value Problems, Academic Press, Orlando, Fl., 1984zbMATHGoogle Scholar
  3. [3]
    Axelsson O., and Gustafsson I., Preconditioning and two-level multigrid methods of arbitrary degree of approximation, Mathematics of Computation 40(1983), 219–242MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Axelsson O. and Maubach J., A time-space finite element discretization technique for the calculation of the electromagnetic field in ferromagnetic materials, Journal for Numerical Methods in Engineering 29(1989), 2085–2111MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Axlesson O. and Maubach J., On the updating and assembly of the hessian matrix in finite element methods, Computer Methods in Applied Mechanics and Engineering 71(1988),41–67MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Axelsson O. and Vassilevski P.S., Algebraic multilevel preconditioning methods I, Numer. Math. 56(1989), 157–177MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Axelsson O. and Vassilevski P.S., Algebraic multilevel preconditiong methods II, Siam J. Numer.Anal. 1990, to appearGoogle Scholar
  8. [8]
    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
  9. [9]
    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
  10. [10]
    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
  11. [11]
    Bank R.E. and Smith R.K., General sparse elimination requires no permanent integer storage, SIAM J. Sci. Stat. Comput. 8(1987), 574–584MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Bank R.E. and Weiser A., Some a posterior error estimators for elliptic partial differential equations, Math. Comp. 44(1985), 283–301MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Becker E.B., Finite Elements, Prentice Hall, Englewood Cliffs, New Jersey, 1981Google Scholar
  14. [14]
    Carey G.F., Barragy R., Mclay R. and Sharma M., Element by element vector and parallel computations, Communications in Applied Numerical Methods 4(1988), 299–307MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    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
  16. [16]
    Ciarlet P.G., The Finite Element Method for Elliptic Problems, North-Holland Publ., Amsterdam, 1978zbMATHGoogle Scholar
  17. [17]
    Deuflhard P., Leinen P. and Yserentant H., Concepts of an adaptive hierarchical finite element code, Preprint SC 88-5 (September 1988), Konrad-Zuse-Zentrum für Informationstechnik, BerlinzbMATHGoogle Scholar
  18. [18]
    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
  19. [19]
    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
  20. [20]
    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
  21. [21]
    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
  22. [22]
    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
  23. [23]
    Rivara M.C., Algorithms for refining triangular grids suitable for adaptive and multigrid techniques, International Journal for Numerical Methods in Engineering 20(1984), 745–756MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Rivara M.C., A grid generator based on 4-triangles conforming mesh refinement algorithms, International Journal for Numerical Methods in Engineering 24(1987), 1343–1354CrossRefzbMATHGoogle Scholar
  25. [25]
    Sewell E.G., Automatic generation of triangulations for piecewise polynomial approximation, Ph.D. Thesis, Purdue University, West Lafayette, IN, 1972Google Scholar
  26. [26]
    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
  27. [27]
    Strang G. and Fix G.S., An analysis of the finite element method, Prentice Hall, Englewood Heights, New Jersey, 1973zbMATHGoogle Scholar
  28. [28]
    Yserentant H., On the multilevel splitting of finite element spaces, Numerische Mathematik 49(1986), 379–412MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Zienkiewicz O., The Finite Element Method in Engineering Science, 3rd edition, Mc Graw-Hill, New York, 1977Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • J. Maubach
    • 1
  1. 1.Faculty of Mathematics and Computer ScienceUniversity of NijmegenNijmegenThe Netherlands

Personalised recommendations