  The aim of the present work is to extend the a priori error for contact
problems with an augmented Lagrangian method. We focus on unilateral contact
problem without friction between an elastic body and a rigid one. We consider
the pushforward of a NURBS space of degree $p$ for the displacement and the
pushforward of a B-Spline space of degree $p-2$ for the Lagrange multipliers.
This specific choice of space is a stable couple of spaces. An optimal a priori
error estimate inspired from the Nitsche's method theory is provided and
compared to the regularity of the solution. We perform a numerical validation
with two- and three-dimensions in small and large deformations with $N2/S0$ and
$N3/S1$ elements.
