A new geometric condition equivalent to the maximum angle condition for tetrahedrons

TL;DR

This paper introduces a new geometric condition that is equivalent to the maximum angle condition for tetrahedrons, aiding in error analysis of Lagrange interpolation, especially on anisotropic tetrahedrons.

Contribution

It presents a novel geometric criterion equivalent to the maximum angle condition, enhancing understanding and analysis of tetrahedral meshes in numerical methods.

Findings

Established an equivalent geometric condition for tetrahedrons.

Facilitated error analysis in Lagrange interpolation.

Applicable to anisotropic tetrahedral meshes.

Abstract

For a tetrahedron, suppose that all internal angles of faces and all dihedral angles are less than a fixed constant $C$ that is smaller than $\pi$. Then, it is said to satisfy the maximum angle condition with the constant $C$. The maximum angle condition is important in the error analysis of Lagrange interpolation on tetrahedrons. This condition ensures that we can obtain an error estimation, even on certain kinds of anisotropic tetrahedrons. In this paper, using two quantities that represent the geometry of tetrahedrons, we present an equivalent geometric condition to the maximum angle condition for tetrahedrons.