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.

翻译：本文提出了一种利用基本凸对偶关系对凸最小化问题进行显式后验误差表示的通用方法。通过利用单元常数向量场空间中的离散正交关系，以及Crouzeix-Raviart元与Raviart-Thomas元之间的离散分部积分公式，所有凸对偶关系都被转移至离散层面，使得原本基于连续论证的显式后验误差表示在数值上变得可行。此外，我们还为原始解提供了一个广义Marini公式，该公式通过给定的离散对偶解确定离散原始解。我们通过Rudin-Osher-Fatemi模型对所有概念进行了基准测试，并由此得到一种自适应算法，该算法实现了（准最优的）线性收敛速率。