Logarithm cannot be removed in maximum norm error estimates for linear finite elements in 3D

TL;DR

This paper demonstrates that for certain 3D finite element meshes, the commonly assumed logarithmic factor in maximum norm error estimates cannot be eliminated, highlighting a fundamental limitation in error analysis.

Contribution

It provides a specific example showing the necessity of the logarithmic factor in maximum norm error bounds for linear finite elements in 3D.

Findings

Logarithmic factor cannot be removed in some 3D cases

Counterexample on tetrahedral mesh in cubic domain

Limits the accuracy expectations of finite element error estimates

Abstract

For linear finite element discretizations of the Laplace equation in three dimensions, we give an example of a tetrahedral mesh in the cubic domain for which the logarithmic factor cannot be removed from the standard upper bounds on the error in the maximum norm.