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

On the DPG method for Signorini problems