On a fixed point study of an inverse problem governed by Stokes equation

Abstract

This work deals with domain decomposition like methods for solving an inverse Cauchy problem governed by Stokes equation. As it is well known, this problem is one of highly ill-posed problems in the Hadamard’s sense (Hadamard 1953 Lectures on Cauchy’s Problem in Linear Partial Differential Equations (New York: Dover)). To solve this problem, we develop a technique based on its reformulation into a fixed point one involving a Steklov like operator. Firstly, we show the existence of its fixed point using the topological degrees of Leray–Schauder tools. Then, we propose a fixed point algorithm allowing us to reproduce the alternating one of Kozlov Mazya (1991 Comput. Math. Math. Phys. 31 45–52). So the proposed approach can be considered as a generalization of Kozlov–Mazya algorithm, since it offers the opportunities to exploit other domain decomposition algorithms for solving this inverse problem. Finally, we investigate the numerical approximation of this problem, using Robin–Robin domain decomposition algorithm and the finite element method. The obtained numerical results show the efficiency of the proposed approach.