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