Conditioning of Finite Volume Element Method for Diffusion Problems with General Simplicial Meshes

TL;DR

This paper analyzes the conditioning of finite volume element discretizations for diffusion problems on arbitrary meshes, providing bounds on the condition number influenced by mesh properties and demonstrating the effectiveness of diagonal scaling.

Contribution

It establishes bounds on the condition number for finite volume element methods on general meshes, considering mesh nonuniformity and scaling effects, with numerical validation.

Findings

Condition number bounds depend on mesh size and nonuniformity.

Diagonal scaling reduces the impact of Euclidean mesh nonuniformity.

Numerical examples confirm theoretical bounds across dimensions.

Abstract

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.