On a regularization approach for solving the inverse Cauchy Stokes problem

Abstract

In this paper, we are interested to an inverse Cauchy problem governed by 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 highly ill-posed problem in the Hadamard's sense [14], 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 [8], for solving an inverse source problem. We show the existence of the regularization optimization problem and prove the convergence of 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 elements method of P 1 − bubble/P 1 type. Finally, we provide some numerical results showing the accuracy, the effectiveness and robustness of the proposed approach.