Conditioning of finite element equations with arbitrary anisotropic meshes

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.

An “immersed” finite element method based on a locally anisotropic remeshing for the incompressible Stokes problem

“…Bounds for the conditioning of the Schur complement are provided, e.g., in [13] or in Proposition 4.47 from [27] as function of the inf-sup constant and the condition number of the pressure mass matrix. A bound for the mass matrix with anisotropic elements is provided, e.g., in [28]. The results are consistent with the theory. )…”

A Fat boundary-type method for localized nonhomogeneous material problems

“…While using fine mesh throughout the whole domain may give an acceptable numerical solution, the computational cost of such an approach may be prohibitive. In contrast, a mesh that is coarse in one region and fine in another, while computationally more attractive, may be difficult to generate and is also known to cause difficulties with preconditioning [5,6,7,13]. Additionally, in some instances the boundary of the subregion may be geometrically complicated, or may change in time, requiring frequent remeshing or the use of complicated local refinement and derefinement techniques [12,15,22,23,24,27].…”

A sketched finite element method for elliptic models

“…A more challenging benchmark test is to consider the FEM solution for a parameter field with non-smooth variation. In this case it is natural to anticipate that any significant jump discontinuities in the profile of p will have an adverse effect on the condition number of the stiffness matrix [13]. For our simulations we choose a piecewise constant approximation of the positive function p(x) .…”