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 \cite{BLO18} (\emph{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.

翻译：我们重新审视著名的Kohn-Vogelius罚方法，并讨论如何将其应用于数据在域内部给定的唯一延拓问题。随后证明，文献\cite{BLO18}（E. Burman, M. Larson, L. Oksanen，《椭圆Cauchy问题的原始对偶混合有限元方法》，SIAM J. Num. Anal., 56 (6), 2018）中针对椭圆Cauchy问题提出的原始对偶混合有限元方法可解释为一种Kohn-Vogelius罚方法，并对其进行修改以适用于利用域内部数据的唯一延拓。我们证明所得线性系统对所有数据均是可逆的。然后表明，通过在离散层面引入奇异摄动Robin条件，可获得充分正则化，从而能够利用条件稳定性推导误差估计。最后展示该方法如何用于利用部分数据识别二阶椭圆算子中的扩散系数。文中给出若干数值算例，展示该方法在唯一延拓及部分数据电阻抗断层成像中的性能。