Volume 47, Issue 2
A Study on the Conditioning of Finite Element Equations with Arbitrary Anisotropic Meshes via a Density Function Approach

Lennard Kamenski & Weizhang Huang

J. Math. Study, 47 (2014), pp. 151-172.

Published online: 2014-06

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  • Abstract

The linear finite element approximation of a general linear diffusion problem with arbitrary anisotropic meshes is considered. The conditioning of the resultant stiffness matrix and the Jacobi preconditioned stiffness matrix is investigated using a density function approach proposed by Fried in 1973. It is shown that the approach can be made mathematically rigorous for general domains and used to develop bounds on the smallest eigenvalue and the condition number that are sharper than existing estimates in one and two dimensions and comparable in three and higher dimensions. The new results reveal that the mesh concentration near the boundary has less influence on the condition number than the mesh concentration in the interior of the domain. This is especially true for the Jacobi preconditioned system where the former has little or almost no influence on the condition number. Numerical examples are presented.

  • Keywords

Conditioning, finite element, anisotropic diffusion, anisotropic mesh, stiffness matrix, extreme eigenvalue, Jacobi preconditioning, diagonal scaling.

  • AMS Subject Headings

65N30, 65N50, 65F35, 65F15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

kamenski@wias-berlin.de (Lennard Kamenski)

whuang@ku.edu (Weizhang Huang)

  • BibTex
  • RIS
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@Article{JMS-47-151, author = {Lennard and Kamenski and kamenski@wias-berlin.de and 13282 and Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany and Lennard Kamenski and Weizhang and Huang and whuang@ku.edu and 12621 and Department of Mathematics, University of Kansas, Lawrence, KS 66045, U.S.A and Weizhang Huang}, title = {A Study on the Conditioning of Finite Element Equations with Arbitrary Anisotropic Meshes via a Density Function Approach}, journal = {Journal of Mathematical Study}, year = {2014}, volume = {47}, number = {2}, pages = {151--172}, abstract = {

The linear finite element approximation of a general linear diffusion problem with arbitrary anisotropic meshes is considered. The conditioning of the resultant stiffness matrix and the Jacobi preconditioned stiffness matrix is investigated using a density function approach proposed by Fried in 1973. It is shown that the approach can be made mathematically rigorous for general domains and used to develop bounds on the smallest eigenvalue and the condition number that are sharper than existing estimates in one and two dimensions and comparable in three and higher dimensions. The new results reveal that the mesh concentration near the boundary has less influence on the condition number than the mesh concentration in the interior of the domain. This is especially true for the Jacobi preconditioned system where the former has little or almost no influence on the condition number. Numerical examples are presented.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v47n2.14.02}, url = {http://global-sci.org/intro/article_detail/jms/9952.html} }
TY - JOUR T1 - A Study on the Conditioning of Finite Element Equations with Arbitrary Anisotropic Meshes via a Density Function Approach AU - Kamenski , Lennard AU - Huang , Weizhang JO - Journal of Mathematical Study VL - 2 SP - 151 EP - 172 PY - 2014 DA - 2014/06 SN - 47 DO - http://doi.org/10.4208/jms.v47n2.14.02 UR - https://global-sci.org/intro/article_detail/jms/9952.html KW - Conditioning, finite element, anisotropic diffusion, anisotropic mesh, stiffness matrix, extreme eigenvalue, Jacobi preconditioning, diagonal scaling. AB -

The linear finite element approximation of a general linear diffusion problem with arbitrary anisotropic meshes is considered. The conditioning of the resultant stiffness matrix and the Jacobi preconditioned stiffness matrix is investigated using a density function approach proposed by Fried in 1973. It is shown that the approach can be made mathematically rigorous for general domains and used to develop bounds on the smallest eigenvalue and the condition number that are sharper than existing estimates in one and two dimensions and comparable in three and higher dimensions. The new results reveal that the mesh concentration near the boundary has less influence on the condition number than the mesh concentration in the interior of the domain. This is especially true for the Jacobi preconditioned system where the former has little or almost no influence on the condition number. Numerical examples are presented.

Lennard Kamenski & Weizhang Huang. (2019). A Study on the Conditioning of Finite Element Equations with Arbitrary Anisotropic Meshes via a Density Function Approach. Journal of Mathematical Study. 47 (2). 151-172. doi:10.4208/jms.v47n2.14.02
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