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

Explicit a posteriori error representation for variational problems and application to TV-minimization