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.

Missing boundary data reconstruction for the heat equation

“…This can be observed by the numerous theoretical and numerical methods introduced to solve this problem. Indeed, several approaches have been investigated for many types of equations, we mention the methods based on the fundamental solutions, we refer to Marin et al, 3 the domain decomposition techniques, we can cite previous works 4,5 and Ben Belgacem and El Fekih, 6 quasi reversibility methods, see Bourgeois and Chesnel, 7 iterative regularizing approaches, see Cemetière et al, 8 another process based on the minimization of the energy‐like functional, we cite the recent paper of Caubet and Dardé 9 and Andrieux et al 10 …”

On numerical resolution of an inverse Cauchy problem modeling the airflow in the bronchial tree

“…Despite this complexity, this kind of problems has a great importance and also still attracting the interest of the scientific community. Indeed, several applications of Cauchy problems governed by different partial differential equations have been treated in previous works: the biharmonic equation in Hadj and Saker (2020) and Lesnic et al (1999), the Helmholtz equation in Qin and Wei (2009), Qin and Wen (2008) and Marin (2010), the Stokes equation in Chakib et al (2018), Chen et al (2005) and Abda et al (2013), heat conduction equation in Onyango et al (2009a, b) and Johansson et al (2011) and elasticity equation in Karageorghis et al (2012), Delvare et al (2010) and Yang and Hsin (2019).…”

“…More precisely, we aim to reconstruct the air velocity and the pressure on the inaccessible boundaries called the artificial boundaries based on the measurements available on the proximal part (Trachea). The approach adopted here is based on domain decomposition method proposed recently in Chakib et al (2018).…”

An efficient D-N alternating algorithm for solving an inverse problem for Helmholtz equation

“…Then it was implemented and improved by relaxation scheme in [21,22,23]. After that, different studies have been done using this algorithm for solving ill-posed problems governed by partial differential equations [2,4,6,7,9,11,12,13]. The KMF alternating procedure for the Helmholtz problem Eqs.…”

“…Recently, this relaxation was accurately investigated to solve inverse Cauchy problem arising in many applications [11,6,30]. In particular this relaxation was used for solving Cauchy problem for modified Helmholtz [3,20,28].…”