  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.
