  In this paper, we are interested to an inverse Cauchy problem governed by the
Stokes equation, called the data completion problem. It consists in determining
the unspecified fluid velocity, or one of its components over a part of its
boundary, by introducing given measurements on its remaining part. As it's
known, this problem is one of the highly ill-posed problems in the Hadamard's
sense \cite{had}, it is then an interesting challenge to carry out a numerical
procedure for approximating their solutions, mostly in the particular case of
noisy data. To solve this problem, we propose here a regularizing approach
based on a coupled complex boundary method, originally proposed in
\cite{source}, for solving an inverse source problem. We show the existence of
the regularization optimization problem and prove the convergence of the
subsequence of optimal solutions of Tikhonov regularization formulations to the
solution of the Cauchy problem. Then we suggest the numerical approximation of
this problem using the adjoint gradient technic and the finite element method
of $P1-bubble/P1$ type. Finally, we provide some numerical results showing the
accuracy, effectiveness, and robustness of the proposed approach.
