  The paper is concerned with a class of nonlinear free boundary problems,
which are usually solved by variational methods based on primal (or
primal-dual) variational settings. We deduce and investigate special relations
(error identities). They show that a certain nonlinear measure of the distance
to the exact solution (specific for each problem) is equivalent to the
respective duality gap, which minimization is a keystone of all variational
numerical methods. Therefore, the identity defines the measure that contains
maximal quantitative information on the quality of a numerical solution
available through these methods. The measure has quadratic terms generated by
the linear part of the differential operator and nonlinear terms associated
with free boundaries. We obtain fully computable two sided bounds of this
measure and show that they provide efficient estimates of the distance between
the minimizer and any function from the corresponding energy space. Several
examples show that for different minimization sequence the balance between
different components of the overall error measure may be different and
domination of nonlinear terms may indicate that coincidence sets are
approximated incorrectly.
