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The nullspace method for the three‐dimensional Stokes problem
Author(s) -
Le Borne Sabine,
Kriemann Ronald
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700207
Subject(s) - basis (linear algebra) , saddle point , computation , saddle , divergence (linguistics) , mathematics , representation (politics) , matrix (chemical analysis) , computer science , exploit , mathematical optimization , algorithm , geometry , linguistics , philosophy , materials science , computer security , politics , political science , law , composite material
Abstract In this paper, we present the preconditioned nullspace method for the iterative solution of the three‐dimensional Stokes problem. In the nullspace method, the original saddle point system is reduced to a positive definite problem by representing the solution with respect to a basis of discretely divergence free vectors. The exact, explicit computation of such a basis typically has non‐optimal (storage and computational) complexity. There exist some algorithms that exploit the sparsity of the matrix and work well for two dimensional problems but fail for three dimensions. Here, we will exploit an implicit representation of the nullspace basis which can be computed efficiently also in a three‐dimensional setting, possibly only as an approximation. We will present some numerical results to illustrate the performance of the resulting solution method. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)