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Conditioning of finite element equations with arbitrary anisotropic meshes


Authors: Lennard Kamenski, Weizhang Huang and Hongguo Xu
Journal: Math. Comp. 83 (2014), 2187-2211
MSC (2010): Primary 65N30, 65N50, 65F35, 65F15
Published electronically: March 5, 2014
MathSciNet review: 3223329
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Abstract: Bounds are developed for the condition number of the linear finite element equations of an anisotropic diffusion problem with arbitrary meshes. They depend on three factors. The first factor is proportional to a power of the number of mesh elements and represents the condition number of the linear finite element equations for the Laplacian operator on a uniform mesh. The other two factors arise from the mesh nonuniformity viewed in the Euclidean metric and in the metric defined by the diffusion matrix. The new bounds reveal that the conditioning of the finite element equations with adaptive anisotropic meshes is much better than what is commonly assumed. Diagonal scaling for the linear system and its effects on the conditioning are also studied. It is shown that the Jacobi preconditioning, which is an optimal diagonal scaling for a symmetric positive definite sparse matrix, can eliminate the effects of mesh nonuniformity viewed in the Euclidean metric and reduce those effects of the mesh viewed in the metric defined by the diffusion matrix. Tight bounds on the extreme eigenvalues of the stiffness and mass matrices are obtained. Numerical examples are given.


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Additional Information

Lennard Kamenski
Affiliation: Department of Mathematics, the University of Kansas, Lawrence, Kansas 66045
Address at time of publication: Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany
Email: kamenski@wias-berlin.de

Weizhang Huang
Affiliation: Department of Mathematics, the University of Kansas, Lawrence, Kansas 66045
Email: whuang@ku.edu

Hongguo Xu
Affiliation: Department of Mathematics, the University of Kansas, Lawrence, Kansas 66045
Email: xu@math.ku.edu

DOI: https://doi.org/10.1090/S0025-5718-2014-02822-6
Keywords: Mesh adaptation, anisotropic mesh, finite element, mass matrix, stiffness matrix, conditioning, extreme eigenvalues, preconditioning, diagonal scaling
Received by editor(s): January 17, 2012
Received by editor(s) in revised form: September 8, 2012, September 17, 2012, and January 4, 2013
Published electronically: March 5, 2014
Additional Notes: This work was supported in part by the DFG (Germany) under grants KA3215/1-1 and KA3215/2-1 and the National Science Foundation (U.S.A.) under grants DMS-0712935 and DMS-1115118.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.