  In this paper, we study affine Deligne--Lusztig varieties $X_w(b)$ when the
finite part of the element $w$ in the Iwahori--Weyl group is a partial
$\sigma$-Coxeter element. We show that such $w$ is a cordial element and
$X_w(b) \neq \emptyset$ if and only if $b$ satisfies a certain Hodge--Newton
indecomposability condition. The main result of this paper is that for such $w$
and $b$, $X_w(b)$ has a simple geometric structure: the $\sigma$-centralizer of
$b$ acts transitively on the set of irreducible components of $X_w(b)$; and
each irreducible component is an iterated fibration over a classical
Deligne--Lusztig variety of Coxeter type, and the iterated fibers are either
$\mathbb A^1$ or $\mathbb G_m$.

Representation Theory;Algebraic Geometry;Number Theory