Singular solutions, graded meshes, and adaptivity for total-variation regularized minimization problems

Recent quasi-optimal error estimates for the finite element approximation of total-variation regularized minimization problems require the existence of a Lipschitz continuous dual solution. We discuss the validity of this condition and devise numerical methods using locally refined meshes that lead to improved convergence rates despite the occurrence of discontinuities. It turns out that nearly linear convergence is possible on suitably constructed meshes.