Correction to: Convergent numerical approximation of the stochastic total variation flow

Springer Science and Business Media LLC - Tập 11 - Trang 1732-1739 - 2022
Ĺubomír Baňas1, Michael Röckner1, André Wilke1
1Department of Mathematics, Bielefeld University, Bielefeld, Germany

Tóm tắt

We correct two errors in our paper [4]. First error concerns the definition of the SVI solution, where a boundary term which arises due to the Dirichlet boundary condition, was not included. The second error concerns the discrete estimate [4,  Lemma 4.4], which involves the discrete Laplace operator. We provide an alternative proof of the estimate in spatial dimension $$d=1$$ by using a mass lumped version of the discrete Laplacian. Hence, after a minor modification of the fully discrete numerical scheme the convergence in $$d=1$$ follows along the lines of the original proof. The convergence proof of the time semi-discrete scheme, which relies on the continuous counterpart of the estimate [4,  Lemma 4.4], remains valid in higher spatial dimension. The convergence of the fully discrete finite element scheme from [4] in any spatial dimension is shown in [3] by using a different approach.

Tài liệu tham khảo

Attouch, H., Buttazzo, G., Michaille, G.: Variational analysis in Sobolev and BV spaces, volume 6 of MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA (2006) Barbu, V., Röckner, M.: Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise. Arch. Ration. Mech. Anal. 209(3), 797–834 (2013) Baňas, Ĺ., Röckner, M., Wilke, A.: Convergent numerical approximation of the stochastic total variation flow: the higher dimensional case. preprint Baňas, Ĺ., Röckner, M., Wilke, A.: Convergent numerical approximation of the stochastic total variation flow. Stoch. Partial Differ. Equ. Anal. Comput. 9(2), 437–471 (2021) Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 2nd edn. Springer, New York (2002) Temam, R.: Problèmes mathématiques en plasticité. Méthodes Mathématiques de l’Informatique [Mathematical Methods of Information Science], vol. 12. Gauthier-Villars, Montrouge (1983)