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Multilevel finite element preconditioning for $ \sqrt{3}$ refinement

Authors: Jan Maes and Peter Oswald
Journal: Math. Comp. 78 (2009), 1869-1890
MSC (2000): Primary 65F10, 65F35, 65N30, 35J20
Published electronically: May 5, 2009
MathSciNet review: 2521270
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Abstract: We develop a BPX-type multilevel method for the numerical solution of second order elliptic equations in $ \mathbb{R}^2$ using piecewise linear polynomials on a sequence of triangulations given by regular $ \sqrt{3}$ refinement. A multilevel splitting of the finest grid space is obtained from the nonnested sequence of spaces on the coarser triangulations using prolongation operators based on simple averaging procedures. The main result is that the condition number of the corresponding BPX preconditioned linear system is uniformly bounded independent of the size of the problem. The motivation to consider $ \sqrt{3}$ refinement stems from the fact that it is a slower topological refinement than the usual dyadic refinement, and that it alternates the orientation of the refined triangles. Therefore we expect a reduction of the amount of work when compared to the classical BPX preconditioner, although both methods have the same asymptotical complexity. Numerical experiments confirm this statement.

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Additional Information

Jan Maes
Affiliation: The first author’s work was done while at the Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3001 Heverlee, Belgium

Peter Oswald
Affiliation: School of Engineering and Science, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany.

Keywords: Multilevel preconditioning, $\sqrt {3}$ subdivision, elliptic equations
Received by editor(s): June 1, 2007
Received by editor(s) in revised form: August 30, 2008
Published electronically: May 5, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.