WIAS Preprint No. 1903, (2013)

A study on the conditioning of finite element equations with arbitrary anisotropic meshes via a density function approach


  • Kamenski, Lennard
  • Huang, Weizhang

2010 Mathematics Subject Classification

  • 65N30 65N50 65F35 65F15


  • conditioning, finite element, anisotropic diffusion, anisotropic mesh, stiffness matrix, extreme eigenvalue, Jacobi preconditioning, diagonal scaling




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.

Appeared in

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

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