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:一个原始对偶混合有限元法,用于扩散系数的反演识别及其与Kohn-Vogelius罚函数方法的关系
翻译后的摘要:
我们重新审视了著名的Kohn-Vogelius罚函数方法,并讨论了如何将其用于唯一延拓问题,在其中,数据是在域的内部给定的。然后,我们展示了用于椭圆型Cauchy问题的原始对偶混合有限元方法 \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})可以被解释为Kohn-Vogelius罚函数方法,并将其修改为允许在公共区域使用数据进行唯一延拓。我们证明了所有数据的结果线性系统都是可逆的。接下来,我们展示了通过在离散级别引入一个奇异扰动的Robin边界条件,可以获得足够的正则化,以使得可以使用条件稳定性来展示误差估计。最后,我们展示了如何将该方法用于具有部分数据的二阶椭圆算子中扩散系数的识别。介绍了一些数值实验,展示了该方法在唯一延拓和部分数据的阻抗计算断层成像中的性能。