  We propose two different Lagrange multiplier methods for contact problems
derived from the augmented Lagrangian variational formulation. Both the
obstacle problem, where a constraint on the solution is imposed in the bulk
domain and the Signorini problem, where a lateral contact condition is imposed
are considered. We consider both continuous and discontinuous approximation
spaces for the Lagrange multiplier. In the latter case the method is unstable
and a penalty on the jump of the multiplier must be applied for stability. We
prove the existence and uniqueness of discrete solutions, best approximation
estimates and convergence estimates that are optimal compared to the regularity
of the solution.
