  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.
