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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
Article

Abstract

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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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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