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

TL;DR

This paper proves the convergence of a global iterative method for fluid-structure interaction problems involving moving geometries, using fixed point theory and domain remapping techniques.

Contribution

It establishes the convergence of an iterative linearization approach for nonlinear fluid-structure interaction models with moving domains.

Findings

The iterative process converges under certain conditions.

Convergence proof uses Banach fixed point theorem.

Remapping onto a fixed domain aids analysis.

Abstract

In this paper a fluid-structure interaction problem for the incompressible Newtonian fluid is studied. We prove the convergence of an iterative process with respect to the computational domain geometry. In our previous works on numerical approximation of similar problems we refer this approach as the global iterative method. 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 argument, where the contractivity of the corresponding mapping is shown due to 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.