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IMA Journal of Numerical Analysis Advance Access originally published online on November 27, 2007
IMA Journal of Numerical Analysis 2008 28(4):888-908; doi:10.1093/imanum/drm031
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© The author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Multigrid methods with Powell–Sabin splines

Hendrik Speleers{dagger}, Paul Dierckx and Stefan Vandewalle

Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3001 Leuven, Belgium

{dagger} Email: hendrik.speleers{at}cs.kuleuven.be. Research assistant of the Fund for Scientific Research Flanders (Belgium).

Received on 27 February 2007. Accepted for publication 27 July 2007.


  Abstract

This paper presents a multigrid algorithm for the solution of the linear systems that arise from a finite-element discretization of second-order elliptic partial differential equations with Powell–Sabin (PS) splines. We show that the method yields a uniform convergence in the l2-norm which is independent of the mesh size. We also briefly consider the use of PS splines for the fourth-order biharmonic problem.

Key Words: Powell–Sabin splines; multigrid; approximation

Dedicated to Prof. M. J. D. Powell on the occasion of his 70th birthday and in honour of his many contributions to numerical analysis.


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