Conditioning of finite element equations with arbitrary anisotropic meshes
American Mathematical Society
Scholarly/refereed, publisher version
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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.
This is the published version, also available here: http://dx.doi.org/10.1090/S0025-5718-2014-02822-6. First published in Math. Comput. in 2014, published by the American Mathematical Society
Kamenski, Lennard., Huang, Weizhang., Xu, Hongguo. "Conditioning of finite element equations with arbitrary anisotropic meshes." Math. Comput. Volume 83, Number 289, September 2014, Pages 2187–2211. http://dx.doi.org/10.1090/S0025-5718-2014-02822-6.
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