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Numerical Algorithms

, Volume 64, Issue 2, pp 349–383 | Cite as

Modified nodal cubic spline collocation for three-dimensional variable coefficient second order partial differential equations

  • Bernard Bialecki
  • Andreas Karageorghis
Original Paper
  • 182 Downloads

Abstract

We formulate a fourth order modified nodal cubic spline collocation scheme for variable coefficient second order partial differential equations in the unit cube subject to nonzero Dirichlet boundary conditions. The approximate solution satisfies a perturbed partial differential equation at the interior nodes of a uniform \(N\times N\times N\) partition of the cube and the partial differential equation at the boundary nodes. In the special case of Poisson’s equation, the resulting linear system is solved by a matrix decomposition algorithm with fast Fourier transforms at a cost \(O(N^3\log N)\). For the general variable coefficient diffusion-dominated case, the system is solved using the preconditioned biconjugate gradient stabilized method.

Keywords

Nodal spline collocation Matrix decomposition algorithm Fast Fourier transforms 

Mathematics Subject Classifications (2010)

65N35 65N22 

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of Applied Mathematics and StatisticsColorado School of MinesGoldenUSA
  2. 2.Department of Mathematics and StatisticsUniversity of CyprusNicosiaCyprus

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