On the convergence of fixed point iterations for the moving geometry in a fluid-structure interaction problem

Abstract

In this paper a fluid-structure interaction problem for incompressible Newtonian fluids is studied. We prove the convergence of an iterative process with respect to the computational domain geometry moving according to a viscoelastic deformation. In our previous joint works on numerical approximation of similar problems we refer to this approach as the global iterative method [1,2]. This iterative approach can be understood as a linearization of the so-called geometric nonlinearity of the underlying model. The proof of the convergence is based on the Banach fixed point Theorem, where the contractivity of the corresponding mapping is shown using the continuous dependence of the weak solution on the given domain deformation. This estimate is obtained by remapping the problem onto a fixed domain and using appropriate divergence-free test functions involving the difference of two solutions.