On the DPG method for Signorini problems

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.