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Recent quasi-optimal error estimates for the finite element approximation oftotal-variation regularized minimization problems using the Crouzeix--Raviartfinite element require the existence of a Lipschitz continuous dual solution,which is not generally given. We provide analytic proofs showing that theLipschitz continuity of a dual solution is not necessary, in general. Using theLipschitz truncation technique, we, in addition, derive error estimates thatdepend directly on the Sobolev regularity of a given dual solution.

Error estimates for total-variation regularized minimization problems with singular dual solutions