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.
翻译:我们纠正了我们文件中的两个错误[4]。第一个错误涉及SVI解决方案的定义,其中不包括由于Drichlet边界条件产生的边界术语。第二个错误涉及离散估计[4, Lemma 4.4],涉及离散拉普尔操作员。我们用一个大片的离散拉平板块块版来提供空间维度估计的替代证据$d=1美元。因此,在对完全离散的数字办法稍作修改之后,以美元=1美元表示的趋同与原始证据的大致相同。时间半分解办法的趋同证明,依赖[4, Lemma 4.4]这一估计的连续对应方,在更高的空间维度中仍然有效。完全离散的有限要素办法与[4]的任何空间维度的趋同在[3]中以不同方法显示。