Conditioning of Finite Element Equations with Arbitrary Anisotropic Meshes

TL;DR

This paper develops bounds for the condition number of finite element equations in anisotropic diffusion problems, showing that adaptive anisotropic meshes can have better conditioning than previously feared, especially with diagonal scaling.

Contribution

It introduces new bounds on the condition number considering mesh anisotropy and demonstrates the effectiveness of diagonal scaling in improving system conditioning.

Findings

Condition number bounds depend on mesh size and anisotropy factors.

Diagonal scaling, especially Jacobi preconditioning, improves conditioning.

Numerical examples confirm theoretical bounds.

Abstract

Bounds are developed for the condition number of the linear finite element equations of an anisotropic diffusion problem with arbitrary meshes. They depend on three factors. The first, factor proportional to a power of the number of mesh elements, represents the condition number of the linear finite element equations for the Laplacian operator on a uniform mesh. The other two factors arise from the mesh nonuniformity viewed in the Euclidean metric and in the metric defined by the diffusion matrix. The new bounds reveal that the conditioning of the finite element equations with adaptive anisotropic meshes is much better than what is commonly feared. Diagonal scaling for the linear system and its effects on the conditioning are also studied. It is shown that the Jacobi preconditioning, which is an optimal diagonal scaling for a symmetric positive definite sparse matrix, can eliminate the effects of mesh nonuniformity viewed in the Euclidean metric and reduce those effects of the mesh viewed in the metric defined by the diffusion matrix. Tight bounds on the extreme eigenvalues of the stiffness and mass matrices are obtained. Numerical examples are given.