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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Conditioning of finite element equations with arbitrary anisotropic meshes
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by Lennard Kamenski, Weizhang Huang and Hongguo Xu PDF
Math. Comp. 83 (2014), 2187-2211 Request permission

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
  • MR Author ID: 326320
  • Email: whuang@ku.edu
  • Hongguo Xu
  • Affiliation: Department of Mathematics, the University of Kansas, Lawrence, Kansas 66045
  • Email: xu@math.ku.edu
  • 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 KA 3215/1-1 and KA 3215/2-1 and the National Science Foundation (U.S.A.) under grants DMS-0712935 and DMS-1115118.
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 83 (2014), 2187-2211
  • MSC (2010): Primary 65N30, 65N50, 65F35, 65F15
  • DOI: https://doi.org/10.1090/S0025-5718-2014-02822-6
  • MathSciNet review: 3223329