A primal dual mixed finite element method for inverse identification of the diffusion coefficient and its relation to the Kohn-Vogelius penalty method
A primal dual mixed finite element method for inverse identification of the diffusion coefficient and its relation to the Kohn-Vogelius penalty method
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DOI：
10.48550/arxiv.2304.10467
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发表时间：
2023-04
期刊：
影响因子：
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通讯作者：
E. Burman
中科院分区：
文献类型：
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作者：
E. Burman
我们重新审视著名的Kohn-Vogelius惩罚方法，并讨论如何使用它的独特的连续性问题的数据是在大部分的域。然后，我们证明了[1]（E。Burman，M.拉森湖李明，李明辉，李明辉，等.椭圆型柯西问题的原-对偶混合有限元方法.分析：56（6），2018）可以被解释为Kohn-Vogelius惩罚方法，并对其进行修改，以允许使用批量数据的唯一延续。我们证明了所得线性系统对于所有数据都是可逆的。然后，我们表明，通过引入一个奇摄动罗宾条件的离散水平上得到充分的正则化，使误差估计可以使用条件稳定性。最后，我们展示了如何使用该方法来识别的扩散系数在一个二阶椭圆算子的部分数据。一些数值例子显示的性能的方法唯一的延续和阻抗计算机断层扫描与部分数据。
We revisit the celebrated Kohn-Vogelius penalty method and discuss how to use it for the unique continuation problem where data is given in the bulk of the domain. We then show that the primal-dual mixed finite element methods for the elliptic Cauchy problem introduced in [1] (E. Burman, M. Larson, L. Oksanen, Primal-dual mixed finite element methods for the elliptic Cauchy problem, SIAM J. Num. Anal., 56(6), 2018) can be interpreted as a Kohn-Vogelius penalty method and modify it to allow for unique continuation using data in the bulk. We prove that the resulting linear system is invertible for all data. Then we show that by introducing a singularly perturbed Robin condition on the discrete level sufficient regularization is obtained so that error estimates can be shown using conditional stability. Finally we show how the method can be used for the identification of the diffusivity coefficient in a second order elliptic operator with partial data. Some numerical examples are presented showing the performance of the method for unique continuation and for impedance computed tomography with partial data.