Recent quasi-optimal error estimates for the finite element approximation of total-variation regularized minimization problems using the Crouzeix--Raviart finite element require the existence of a Lipschitz continuous dual solution, which is not generally given. We provide analytic proofs showing that the Lipschitz continuity of a dual solution is not necessary, in general. Using the Lipschitz truncation technique, we, in addition, derive error estimates that depend directly on the Sobolev regularity of a given dual solution.
翻译:使用Crouzix-Raviart有限要素对总变差固定最小化问题的有限要素近似近似值的最近近似最佳误差估计要求存在一个Lipschitz连续的双重解决办法,但通常没有提供这种解决办法。我们提供分析证明,表明通常没有必要使用Lipschitz双重解决办法的连续性。此外,我们利用Lipschitz截断技术,得出直接取决于特定双重解决办法的索博列斯常规性的误差估计。