BIT Numerical Mathematics

, Volume 37, Issue 4, pp 781–803 | Cite as

Multigrid and multilevel methods for quadratic spline collocation

  • Christina C. Christara
  • Barry Smith


Multigrid methods are developed and analyzed for quadratic spline collocation equations arising from the discretization of one-dimensional second-order differential equations. The rate of convergence of the two-grid method integrated with a damped Richardson relaxation scheme as smoother is shown to be faster than 1/2, independently of the step-size. The additive multilevel versions of the algorithms are also analyzed. The development of quadratic spline collocation multigrid methods is extended to two-dimensional elliptic partial differential equations. Multigrid methods for quadratic spline collocation methods are not straightforward: because the basis functions used with quadratic spline collocation are not nodal basis functions, the design of efficient restriction and extension operators is nontrivial. Experimental results, with V-cycle and full multigrid, indicate that suitably chosen multigrid iteration is a very efficient solver for the quadratic spline collocation equations.

AMS subject classification

65N22 65N35 65N55 65F10 

Key words

Elliptic PDEs high order splines iterative solvers multigrid preconditioning 


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  1. 1.
    A. Brandt,Multi-level adaptive solutions to boundary-value problems, Math. Comp., 31:138 (1977), pp. 333–390.zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    C. C. Christara,Spline collocation methods, software and architectures for linear elliptic boundary value problems, Ph.D. thesis, Purdue University, IN, 1988.Google Scholar
  3. 3.
    C. C. Christara,Schur complement preconditioned conjugate gradient methods for spline collocation equations, in Proceedings of the 1990 International Conference on Supercomputing (ICS90), June 1990, Amsterdam, the Netherlands, sponsored by ACM, pp. 108–120.Google Scholar
  4. 4.
    C. C. Christara,Quadratic spline collocation methods for elliptic partial differential equations, BIT, 34:1 (1994), pp. 33–61.zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    C. C. Christara,Parallel solvers for spline collocation equations, Advances in Engineering Software, 27:1/2 (1996), pp. 71–89.CrossRefGoogle Scholar
  6. 6.
    J. Gary,The multigrid iteration applied to the collocation method, SIAM J. Numer. Anal., 18:2 (1981), pp. 211–224.zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    W. D. Gropp, and B. Smith,Users manual for KSP: Data-structure-neutral codes implementing Krylov space methods, Tech. Rep. ANL-93/30, Argonne National Laboratory, 1993.Google Scholar
  8. 8.
    W. Hackbusch,Multi-Grid Methods and Applications, Springer Verlag, 1980.Google Scholar
  9. 9.
    W. Hackbusch and U. Trottenberg (eds.),Multi-Grid Methods, Lecture Notes in Mathematics 960, Springer Verlag, 1982.Google Scholar
  10. 10.
    W. Hackbusch,Iterative Solution of Large Sparse Systems of Equations, Applied Mathematical Sciences 95, Springer Verlag, 1994.Google Scholar
  11. 11.
    E. N. Houstis, J. R. Rice, C. C. Christara, and E. A. Vavalis,Performance of scientific software, Mathematical Aspects of Scientific Software, J. R. Rice, ed., Springer Verlag, 1988, pp. 123–156.Google Scholar
  12. 12.
    J. Xu,A new class of iterative methods for nonselfadjoint or indefinite problems, SIAM J. Numer. Anal., 29:2 (1992), pp. 303–319.zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    H. Yserentant,Preconditioning indefinite discretization matrices, Numer. Math., 54 (1988), pp. 719–734.MathSciNetCrossRefGoogle Scholar
  14. 14.
    H. Yserentant,Old and new convergence proofs for multigrid methods, Acta Numerica, 1993, pp. 285–326.Google Scholar

Copyright information

© Swets & Zeitlinger 1997

Authors and Affiliations

  • Christina C. Christara
    • 1
  • Barry Smith
    • 2
  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada
  2. 2.Mathematics and Computer Science DivisionArgonne National LaboratoryArgonneUSA

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