  We analyse the subring of the Chow ring with support generated by the
irreducible components of the special fibre of the Gross-Schoen
desingularization of a d-fold self product of a semi-stable curve over the
spectrum of a discrete valuation ring. For this purpose we develop a calculus
which allows to determine intersection numbers in the special fibre explicitly.
As input our simplicial calculus needs only combinatorial data of the special
fibre. It yields a practical procedure for calculating even self intersections
in the special fibre. The first ingredient of our simplicial calculus is a
localization formula, which reduces the problem of calculating intersection
numbers to a special situation. In order to illustrate how our simplicial
calculus works, we calculate all intersection numbers between divisors with
support in the special fibre in dimension three and four. The localization
formula and the general idea were already presented for d=2 in a paper of
Shou-Wu Zhang. In our present work we achieve a generalisation to arbitrary d.
