On a regularization approach for solving the inverse Cauchy Stokes problem

TL;DR

This paper introduces a regularization method using a coupled complex boundary approach to solve the highly ill-posed inverse Cauchy problem for the Stokes equation, demonstrating convergence and numerical effectiveness.

Contribution

It proposes a novel regularization technique based on a coupled complex boundary method for the inverse Stokes problem, with proven convergence and numerical validation.

Findings

The regularization approach converges to the true solution.

Numerical results confirm the method's accuracy and robustness.

The method effectively handles noisy data in inverse problems.

Abstract

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.