  We present a framework for analyzing non-linear $\mathbb{R}^d$-valued
subdivision schemes which are geometric in the sense that they commute with
similarities in $\mathbb{R}^d$. It admits to establish
$C^{1,\alpha}$-regularity for arbitrary schemes of this type, and
$C^{2,\alpha}$-regularity for an important subset thereof, which includes all
real-valued schemes. Our results are constructive in the sense that they can be
verified explicitly for any scheme and any given set of initial data by a
universal procedure. This procedure can be executed automatically and
rigorously by a computer when using interval arithmetics.

Metric Geometry;Classical Analysis and ODEs