A Study on the Conditioning of Finite Element Equations with Arbitrary Anisotropic Meshes via a Density Function Approach
Year: 2014
Author: Lennard Kamenski, Weizhang Huang
Journal of Mathematical Study, Vol. 47 (2014), Iss. 2 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jms.v47n2.14.02
Journal of Mathematical Study, Vol. 47 (2014), Iss. 2 : pp. 151–172
Published online: 2014-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 22
Keywords: Conditioning finite element anisotropic diffusion anisotropic mesh stiffness matrix extreme eigenvalue Jacobi preconditioning diagonal scaling.