Quasi-uniform triangulations and discretizations

Question

Given an interval $\Omega=(0,1)$ I can define a family of discretizations (triangulations) $\{\tau_n\}_{n\in\mathbb{N}}$ where, for a fixed $\bar{n}\in\mathbb{N}$, we have a certain sequence of $\bar{n}+2$ points $x_0<\dots<x_{\bar{n}+1}$, so actually $\bar{n}$ intervals.

Now my question is, when is my family of discretizations quasi-uniform?

I have seen the definition that it is if $\exists c>0$ such that $\frac{h_{max}^{(n)}}{h_{min}^{(n)}}<c$ where $h_{min}^{(n)},h_{max}^{(n)}$ are respectively the minimum and the maximum lengths of the intervals in the discretization $\tau_n$, and $c$ shouldn't depend on $n$.

On another book I've seen another definition, which simply compares the minimum diameter of the triangulation $\tau_n$ with the length of an interval in a uniform triangulation of $n+2$ points on $\Omega$, which is $\frac{1}{n+1}$. In formulas, this writes:

$\frac{h_{min}^{(n)}}{1/(n+1)}>\varepsilon$

for every $n$, where $\varepsilon$ should not depend on $n$. So in case such an $\varepsilon$ exists, we say the family of discretizations quasi-uniform.

So my questions are:

1. Which one of these definitions is correct?

2. If both are, are they equivalent?

3. Is the notion of quasi-uniform discretization of a 1-dimensional domain related to the one of regularity (or shape-regularity) of a triangulation of a 2-dimensional polygonal domain?

Answer 1

The first implies the second: assume the first is satisfied, then $h_\max\ge \frac1{n+1}$, and the second inequality is implied.

The second does not imply the first: choose $\epsilon\in (0,1)$. Let $n$ intervals be of length $\frac\epsilon{n+1}$, then the remaining (largest) element is of size $$ h_\max = 1- \epsilon \frac n{n+1}. $$ Then we have for the ratio $$ \frac{h_\max}{h_\min} = \frac{n+1-\epsilon n}{n+1} \cdot \frac{n+1}\epsilon = \epsilon^{-1}( (1-\epsilon)n +1 ) \to \infty $$ for $n\to \infty$.

Shape regularity is a weaker property. It means that the diameter of the largest ball contained in one element of the same order as the diameter of the element. It is trivially fulfilled for 1d-problems.