Numerische Mathematik

, Volume 52, Issue 4, pp 427–458 | Cite as

The hierarchical basis multigrid method

  • Randolph E. Bank
  • Todd F. Dupont
  • Harry Yserentant
Article

Summary

We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauß-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigridV-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.

Subject Classifications

AMS(MOS): 65F10 65F35 65N20 65N30 CR:G1.8 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Axelsson, O., Barker, V.A.: Finite Element Solution of Boundary Value Problems: Theory and Computation. New York: Academic Press 1984Google Scholar
  2. 2.
    Bank, R.E.: PLTMG User's Guide, Edition 4.0; Technical Report, Department of Mathematics. University of California at San Diego 1985Google Scholar
  3. 3.
    Bank, R.E., Dupont, T.: Analysis of a Two-Level Scheme for Solving Finite Element Equations. Report CNA-159; Center for Numerical Analysis, University of Texas at Austin 1980Google Scholar
  4. 4.
    Bank, R.E., Dupont, T.: An Optimal Order Process for Solving Finite Element Equations. Math. Comput.36, 35–51 (1981)Google Scholar
  5. 5.
    Bank, R.E., Sherman, A., Weiser, A.: Refinement Algorithms and Data Structures for Local Mesh Refinement. In: Scientific Computing (R. Stepleman et al. eds.), Amsterdam: IMACS/North Holland 1983Google Scholar
  6. 6.
    Braess, D.: The Contraction Number of a Multigrid Method for Solving the Poisson Equation. Numer. Math.37, 387–404 (1981)Google Scholar
  7. 7.
    Bramble, J.H., Pasciak, J.E., Schatz, A.H.: An Iterative, Method for Elliptic Problems and Regions Partioned into Substructures. Math. Comput.46, 361–369 (1986)Google Scholar
  8. 8.
    Bramble, J.H., Pasciak, J.E., Schatz, A.H.: The Construction of Preconditioners for Elliptic Problems by Substructuring. I. Math. Comput.47, 103–134 (1986)Google Scholar
  9. 9.
    Hackbusch, W.: Multigrid Methods and Applications. Berlin Heidelberg New York: Springer 1985Google Scholar
  10. 10.
    Hageman, L.A., Young, D.M.: Applied Iterative Methods. New York: Academic Press 1981Google Scholar
  11. 11.
    Maitre, J.F., Musy, F.: The Contraction Number of a Class of Two-Level Methods; an Exact Evaluation for some Finite Element Subspaces and Model Problems. In: Multigrid Methods (W. Hackbusch, U. Trottenberg eds.), Lect. Notes Math. 960, Berlin Heidelberg New York: Springer 1982Google Scholar
  12. 12.
    Rivara, M.C.: Algorithms for Refining Triangular Grids Suitable for Adaptive and Multigrid Techniques. Int J. Numer. Methods Eng.20, 745–756 (1984)Google Scholar
  13. 13.
    Young, D.M.: Convergence Properties of the Symmetric and Unsymmetric Successive Overrelaxation Method and Related Methods. Math. Comput.24, 793–807 (1970)Google Scholar
  14. 14.
    Yserentant, H.: On the Multi-Level Splitting of Finite Element Spaces. Numer. Math.49, 379–412 (1986)Google Scholar
  15. 15.
    Yserentant, H.: Hierarchical Bases Give Conjugate Gradient Type Methods a Multigrid Speed of Convergence. Appl. Math. Comput.19, 347–358 (1986)Google Scholar
  16. 16.
    Yserentant, H.: On the Multi-Level Splitting of Finite Element Spaces for Indefinite Elliptic Boundary Value Problems. Siam J. Numer. Anal.23, 581–595 (1986)Google Scholar
  17. 17.
    Yserentant, H.: Hierarchical Bases of Finite Element Spaces in the Discretization of Nonsymmetric Elliptic Boundary Value Problems. Computing35, 39–49 (1985)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Randolph E. Bank
    • 1
  • Todd F. Dupont
    • 2
  • Harry Yserentant
    • 3
  1. 1.Department of MathematicsUniversity of California at San DiegoLa JollaUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA
  3. 3.Fachbereich MathematikUniversität DortmundDortmund 50Federal Republic of Germany

Personalised recommendations