Mathematics – Numerical Analysis
Scientific paper
2012-01-17
Mathematics
Numerical Analysis
26 pages, 7 figures
Scientific paper
Bounds are developed for the condition number of the linear system resulting from the finite element discretization of an anisotropic diffusion problem with arbitrary meshes. These bounds are shown to depend on three major factors: a factor representing the base order corresponding to the condition number for a uniform mesh, a factor representing the effects of the mesh M-nonuniformity (mesh nonuniformity in the metric tensor defined by the diffusion matrix), and a factor representing the effects of the mesh volume-nonuniformity. Diagonal scaling for the finite element linear system and its effects on the conditioning are studied. It is shown that a properly chosen diagonal scaling can eliminate the effects of the mesh volume-nonuniformity and reduce the effects of the mesh M-nouniformity on the conditioning of the stiffness matrix. In particular, the bound after a proper diagonal scaling depends only on a volume-weighted average (instead of the maximum for the unscaled case) of a quantity measuring the mesh M-nonuniformity. Bounds on the extreme eigenvalues of the stiffness and mass matrices are also investigated. Numerical examples are presented to verify the theoretical findings.
Huang Weizhang
Kamenski Lennard
Xu Hongguo
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