Estimates of the distance to exact solutions for a class of free boundary problems

We consider elliptic and parabolic variational inequalities generated by problems with obstacles. Existence and uniqueness of generalized solutions, their regularity and local properties are well studied.
However, with rare exceptions, exact solutions are unknown. Hence, in the overwhelming majority of cases, we are forced to consider a certain approximation instead of the exact solution. Therefore inevitably there rises the question: How to verify that it is indeed close to the solution? Moreover, for the considered class of problems there is a harder question: Can we trust in approximations of free boundaries computed by standard numerical approaches?
The key point of analysis is the a posteriori error identity, which relates a certain measure of the distance with a computable complex that depend on approximations and known data only. Such identities are known for a wide class of variational problems. We discuss them and corresponding estimates for the classical obstacle problem, problems with thin obstacles, and parabolic obstacle problems.