The maximum angle condition is not necessary for convergence of the finite element method

Abstract

We show that the famous maximum angle condition in the finite element analysis is not necessary to achieve the optimal convergence rate when simplicial finite elements are used to solve elliptic problems. This condition is only sufficient. In fact, finite element approximations may converge even though some dihedral angles of simplicial elements tend to π .

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

“…Again the condition is sufficient to guarantee the optimal convergence of the finite element method. However, it has been noted in [24] that the maximum angle condition is not necessary and the finite element method for 2D problems may converge optimally without a maximum angle condition satisfied. Moreover, since we consider a saddle point problem, an inf-sup condition has to be satisfied as well, and finite element schemes that are stable on well shaped elements may not be stable on anisotropic ones.…”

Divergence of fem: Babuška-Aziz triangulations revisited

“…The most widely known sufficient condition, the maximum angle condition, goes back to Babuška and Aziz [2], and Jamet [6]. The fact that the maximum angle condition is not necessary for the convergence of the FEM was demonstrated in [5]. For approaches to proving FEM convergence and error estimates for certain classes of anisotropic partitions based on other local interpolation operators, see [1].…”

On the generalization of the Synge–Křížek maximum angle condition for d-simplices

“…Remark 2.8. It is worth to mention that the maximum angle condition is only sufficient to provide the convergence of the finite element approximations as shown in [10].…”