  We develop a discrete counterpart of the De Giorgi-Nash-Moser theory, which
provides uniform H\"older-norm bounds on continuous piecewise affine finite
element approximations of second-order linear elliptic problems of the form
$-\nabla \cdot(A\nabla u)=f-\nabla\cdot F$ with $A\in
L^\infty(\Omega;\mathbb{R}^{n\times n})$ a uniformly elliptic matrix-valued
function, $f\in L^{q}(\Omega)$, $F\in L^p(\Omega;\mathbb{R}^n)$, with $p > n$
and $q > n/2$, on $A$-nonobtuse shape-regular triangulations, which are not
required to be quasi-uniform, of a bounded polyhedral Lipschitz domain $\Omega
\subset \mathbb{R}^n$.
