On the circumradius condition for piecewise linear triangular elements

TL;DR

This paper emphasizes the importance of the circumradius condition over the maximum angle condition for finite element convergence, supported by numerical experiments and its relation to surface area.

Contribution

It establishes the circumradius condition as a more critical criterion than the maximum angle condition for finite element error convergence.

Findings

Circumradius condition is essential for finite element convergence.

Numerical experiments confirm the optimality of the circumradius condition.

The circumradius condition relates closely to surface area definition.

Abstract

We discuss the error analysis of linear interpolation on triangular elements. We claim that the circumradius condition is more essential then the well-known maximum angle condition for convergence of the finite element method. Numerical experiments show that this condition is the best possible. We also point out that the circumradius condition is closely related to the definition of surface area.