  We consider primal-dual mixed finite element methods for the solution of the
elliptic Cauchy problem, or other related data assimilation problems. The
method has a local conservation property. We derive a priori error estimates
using known conditional stability estimates and determine the minimal amount of
weakly consistent stabilization and Tikhonov regularization that yields optimal
convergence for smooth exact solutions. The effect of perturbations in data is
also accounted for. A reduced version of the method, obtained by choosing a
special stabilization of the dual variable, can be viewed as a variant of the
least squares mixed finite element method introduced by Dard\'e, Hannukainen
and Hyv\"onen in \emph{An {$H\sb {\sf{div}}$}-based mixed quasi-reversibility
method for solving elliptic {C}auchy problems}, SIAM J. Numer. Anal., 51(4)
2013. The main difference is that our choice of regularization does not depend
on auxiliary parameters, the mesh size being the only asymptotic parameter.
Finally, we show that the reduced method can be used for defect correction
iteration to determine the solution of the full method. The theory is
illustrated by some numerical examples.
