BIT Numerical Mathematics

, Volume 34, Issue 1, pp 33–61 | Cite as

Quadratic spline collocation methods for elliptic partial differential equations

  • Christina C. Christara


We consider Quadratic Spline Collocation (QSC) methods for linear second order elliptic Partial Differential Equations (PDEs). The standard formulation of these methods leads to non-optimal approximations. In order to derive optimal QSC approximations, high order perturbations of the PDE problem are generated. These perturbations can be applied either to the PDE problem operators or to the right sides, thus leading to two different formulations of optimal QSC methods. The convergence properties of the QSC methods are studied. OptimalO(h3−j) global error estimates for thejth partial derivative are obtained for a certain class of problems. Moreover,O(h4−j) error bounds for thejth partial derivative are obtained at certain sets of points. Results from numerical experiments verify the theoretical behaviour of the QSC methods. Performance results also show that the QSC methods are very effective from the computational point of view. They have been implemented efficiently on parallel machines.

AMS(MOS) subject classifications

65N35 65N15 

Key words

spline collocation elliptic partial differential equations second order boundary value problems 


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Copyright information

© the BIT Foundation 1994

Authors and Affiliations

  • Christina C. Christara
    • 1
  1. 1.Department of Computer SciencesUniversity of TorontoTorontoCanada

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