Numerische Mathematik

, Volume 58, Issue 1, pp 163–184 | Cite as

Two preconditioners based on the multi-level splitting of finite element spaces

  • Harry Yserentant


The hierarchical basis preconditioner and the recent preconditioner of Bramble, Pasciak and Xu are derived and analyzed within a joint framework. This discussion elucidates the close relationship between both methods. Special care is devoted to highly nonuniform meshes; exclusively local properties like the shape regularity of the finite elements are utilized.

Subject classifications

AMS(MOS) 65F10 65F35 65N20 65N30 CR: G 1.8 


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  1. 1.
    Bank, R.E.: PLTMG: A software package for solving elliptic partial differential equations. Philadelphia: SIAM 1990Google Scholar
  2. 2.
    Bank, R.E., Dupont, T., Yserentant, H.: The hierarchical basis multigrid method. Numer. Math.52, 427–458 (1988)Google Scholar
  3. 3.
    Bank, R.E., Sherman, A.H., Weiser, A.: Refinement algorithms and data structures for regular local mesh refinement. In: Stepleman, R. (eds.) Scientific computing, pp. 3–17. Amsterdam: IMACS/North Holland 1983Google Scholar
  4. 4.
    Bramble, J.H., Pasciak, J.E., Xu, J.: Parallel multilevel preconditioners. Math. Comput. (to appear)Google Scholar
  5. 5.
    Crouzeix, M., Thomée, V.: The stability inL p andW 1p of theL 2-projection onto finiteelement function spaces. Math. Comput.48, 521–532 (1987)Google Scholar
  6. 6.
    Deuflhard, P., Leinen, P., Yserentant, H.: Concepts of an adaptive hierarchical finite element code. IMPACT of Computing in Science and Engineering1, 3–35 (1989)Google Scholar
  7. 7.
    Hackbusch, W.: Multigrid methods and applications. Berlin, Heidelberg, New York: Springer 1985Google Scholar
  8. 8.
    Hackbusch, W.: Theorie und Numerik elliptischer Differentialgleichungen. Stuttgart: Teubner 1986 (English translation in preparation)Google Scholar
  9. 9.
    Ong, M.E.G.: Hierarchical basis preconditioners for second order elliptic problems in three dimensions. Technical Report No. 89-3, Department of Applied Mathematics, University of Washington, Seattle 1989Google Scholar
  10. 10.
    Oswald, P.: On estimates for hierarchic basis representations of finite element functions. Technical Report N/89/16, Sektion Mathematik, Friedrich-Schiller Universität Jena 1989Google Scholar
  11. 11.
    Xu, J.: Theory of multilevel methods. Report No. AM48, Department of Mathematics. Pennsylvania State University 1989Google Scholar
  12. 12.
    Yserentant, H.: On the multi-level splitting of finite element spaces. Numer. Math.49, 379–412 (1986)Google Scholar
  13. 13.
    Yserentant, H.: Hierarchical bases give conjugate gradient type methods a multigrid speed of convergence. Applied Mathematics and Computation19, 347–358 (1986)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Harry Yserentant
    • 1
  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany

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