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

TL;DR

This paper rigorously analyzes how arbitrary anisotropic meshes affect the conditioning of finite element equations, providing sharper bounds and revealing that interior mesh concentration impacts conditioning more than boundary concentration.

Contribution

It develops a mathematically rigorous density function approach to bound eigenvalues and condition numbers for anisotropic meshes in finite element methods, improving upon existing estimates.

Findings

Interior mesh concentration affects condition number more than boundary concentration.

Jacobi preconditioning reduces the influence of boundary mesh concentration.

New bounds are sharper in 1D and 2D, comparable in higher dimensions.

Abstract

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.