Computations on preconditioning cubic spline collocation method of elliptic equations

  • Yong Hun Lee


In this work we investigate the finite element preconditioning method for theC 1-cubic spline collocation discretizations for an elliptic operatorA defined byAu:=−Δu+a 1 u x +a 2 u y +a 0 u in the unit square with some boundary conditions. We compute the condition number and the numerical solution of the preconditioning system for the several example problems. Finally, we compare the this preconditioning system with the another preconditioning system.

AMS Mathematics Subject Classification

65N30 65N35 65F05 65F10 

Key words and phrases

preconditioned matrix cubic spline collocation finite element method 


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  1. 1.
    C. de Boor;Spline tool box for use with Matlab, The Mathworks Inc., 1996.Google Scholar
  2. 2.
    J. Cerutti and S. V. Parter;Collocation Methods for Parabolic Partial Differential Equations in one dimensional space, Numer. Math.,26 (1974), 227–254.CrossRefMathSciNetGoogle Scholar
  3. 3.
    M. O. Deville and E. H. Mund;Finite-Element Preconditioning for Pseudospectral Solutions of Elliptic Problems, SIAM J. Sci. Stat. Comput.11 (1990), 311–342.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    J. Douglas and T. Dupont;Collocation Methods for Parabolic Equations in a Single Space Variable, Lecture Notes in Mathematics385, Springer-Verlag. (1974).Google Scholar
  5. 5.
    H. O. Kim, S. D. Kim and Y. H. Lee;Finite difference preconditioning cubic spline collocation method of elliptic equations, Numer. Math.,77 (1997), 83–103.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    S. D. Kim and Y. H. Lee;Analysis on Eigenvalues for preconditioning cubic spline collocation method of elliptic equations, Lin. Anal. and Its Appl., to appear.Google Scholar
  7. 7.
    S. D. Kim and S. V. Parter;Preconditioning cubic spline collocation discretization of elliptic equations, Numer. Math.,72 (1995), 39–72.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    S. A. Orszag;Spectral methods for problems in complex geometries, J. Comp. Physics,37 (1980), 70–92.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Y. Saad and M. H. SchultzGMRES: A generalized minimal residual algorithm for solving non-symmetric linear systems, SIAM J. Sci. Stat. Comput.7 (1986), 856–869.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    H. A. Van der Vorst;Bi-CGSTAB: A fast and smoothly converging variant of BiCG for the solution of nonsymmetric linear system, SIAM J. Sci. Stat. Comput.,13 (1992), 631–644.MATHCrossRefGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2001

Authors and Affiliations

  1. 1.Department of MathematicsChonbuk National University, ChonjuChonbukKorea

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