  We derive and analyze discontinuous Petrov-Galerkin methods with optimal test
functions for Signorini-type problems as a prototype of a variational
inequality of the first kind. We present different symmetric and non-symmetric
formulations where optimal test functions are only used for the PDE part of the
problem, not the boundary conditions. For the symmetric case and lowest order
approximations, we provide a simple a posteriori error estimate. In a second
part, we apply our technique to the singularly perturbed case of reaction
dominated diffusion. Numerical results show the performance of our method and,
in particular, its robustness in the singularly perturbed case.
