  The conditioning of the linear finite volume element discretization for
general diffusion equations is studied on arbitrary simplicial meshes. The
condition number is defined as the ratio of the maximal singular value of the
stiffness matrix to the minimal eigenvalue of its symmetric part. This
definition is motivated by the fact that the convergence rate of the
generalized minimal residual method for the corresponding linear systems is
determined by the ratio. An upper bound for the ratio is established by
developing an upper bound for the maximal singular value and a lower bound for
the minimal eigenvalue of the symmetric part. It is shown that the bound
depends on three factors, the number of the elements in the mesh, the mesh
nonuniformity measured in the Euclidean metric, and the mesh nonuniformity
measured in the metric specified by the inverse diffusion matrix. It is also
shown that the diagonal scaling can effectively eliminates the effects from the
mesh nonuniformity measured in the Euclidean metric. Numerical results for a
selection of examples in one, two, and three dimensions are presented.
