  Gradient schemes is a framework which enables the unified convergence
analysis of many different methods -- such as finite elements (conforming,
non-conforming and mixed) and finite volumes methods -- for $2^{\rm nd}$ order
diffusion equations. We show in this work that the gradient schemes framework
can be extended to variational inequalities involving mixed Dirichlet, Neumann
and Signorini boundary conditions. This extension allows us to provide error
estimates for numerical approximations of such models, recovering known
convergence rates for some methods, and establishing new convergence rates for
schemes not previously studied for variational inequalities. The general
framework we develop also enables us to design a new numerical method for the
obstacle and Signorini problems, based on hybrid mimetic mixed schemes. We
provide numerical results that demonstrate the accuracy of these schemes, and
confirm our theoretical rates of convergence.
