  In this paper, we propose a general approach for explicit a posteriori error
representation for convex minimization problems using basic convex duality
relations. Exploiting discrete orthogonality relations in the space of
element-wise constant vector fields as well as a discrete integration-by-parts
formula between the Crouzeix-Raviart and the Raviart-Thomas element, all convex
duality relations are transferred to a discrete level, making the explicit a
posteriori error representation -- initially based on continuous arguments only
-- practicable from a numerical point of view. In addition, we provide a
generalized Marini formula for the primal solution that determines a discrete
primal solution in terms of a given discrete dual solution. We benchmark all
these concepts via the Rudin-Osher-Fatemi model. This leads to an adaptive
algorithm that yields a (quasi-optimal) linear convergence rate.
