  Variational inequalities play in many applications an important role and are
an active research area. Optimal a priori error estimates in the natural energy
norm do exist but only very few results in other norms exist. Here we consider
as prototype a simple Signorini problem and provide new optimal order a priori
error estimates for the trace and the flux on the Signorini boundary. The a
priori analysis is based on the exact and a mesh-dependent Steklov-Poincar\'e
operator as well as on duality in Aubin-Nitsche type arguments. Numerical
results illustrate the convergence rates of the finite element approach.
