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We revisit the celebrated Kohn-Vogelius penalty method and discuss how to useit for the unique continuation problem where data is given in the bulk of thedomain. We then show that the primal-dual mixed finite element methods for theelliptic 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 aKohn-Vogelius penalty method and modify it to allow for unique continuationusing data in the bulk. We prove that the resulting linear system is invertiblefor all data. Then we show that by introducing a singularly perturbed Robincondition on the discrete level sufficient regularization is obtained so thaterror estimates can be shown using conditional stability.  Finally we show how the method can be used for the identification of thediffusivity coefficient in a second order elliptic operator with partial data.Some numerical examples are presented showing the performance of the method forunique continuation and for impedance computed tomography with partial data.

A primal dual mixed finite element method for inverse identification of  the diffusion coefficient and its relation to the Kohn-Vogelius penalty  method