On angle conditions in the finite element method

Abstract

Angle conditions play an important role in the analysis of the finite element method. They enable us to derive the optimal interpolation order and prove convergence of this method, to derive various a posteriori error estimates, to perform regular mesh refinements, etc. In 1968, Miloˇs Zl´amal introduced the minimum angle condition for triangular elements. From that time onward many other useful geometric angle conditions on the shape of elements appeared. In this paper, we shall give a survey of various generalizations of the minimum and also maximum angle condition in the finite element method and present some of their applications.

Key words: simplicial finite elements, minimum and maximum angle condition, ball conditions, acute and nonobtuse partitions, two-sided error estimates, a priori error estimates, discrete maximum principle.

AMS subject classifications: 65N30, 65N50, 52B11, 51M20