Full Discretization of Cauchy's Problem by Lavrentiev--Finite Element Method

We conduct a detailed study of the fully discrete finite element approximation of the data completion problem. This is the continuation of [ Numer. Math., 139 (2016), pp. 1--25], where the variational problem, resulting from the Kohn--Vogelius duplication framed into the Steklov--Poincaré condensation approach, was semidiscretized. Under the condition that the problem has a solution, we derive a bound of the error with respect to the mesh-size and the Lavrentiev regularization parameter. Sharp local finite element estimates, such as those derived by Nitsche and Schatz [ Math. Comp., 28 (1974), pp. 937--958], are the central technical tools of the analysis.