Error Analysis of Lagrange Interpolation on Tetrahedrons

TL;DR

This paper introduces a new error estimation method for Lagrange interpolation on tetrahedrons that does not require geometric restrictions, making it applicable to any tetrahedral mesh including irregular shapes.

Contribution

The paper provides a novel error bound based on diameter and projected circumradius, removing the need for shape regularity assumptions.

Findings

Error bounds depend on diameter and projected circumradius

Applicable to arbitrary tetrahedral meshes including thin tetrahedrons

No geometric restrictions needed for error estimation

Abstract

This paper describes the analysis of Lagrange interpolation errors on tetrahedrons. In many textbooks, the error analysis of Lagrange interpolation is conducted under geometric assumptions such as shape regularity or the (generalized) maximum angle condition. In this paper, we present a new estimation in which the error is bounded in terms of the diameter and projected circumradius of the tetrahedron. Because we do not impose any geometric restrictions on the tetrahedron itself, our error estimation may be applied to any tetrahedralizations of domains including very thin tetrahedrons.