# On the convergence rate of the Dirichlet-Neumann iteration for unsteady thermal fluid structure interaction

## TL;DR

This paper analyzes the convergence rate of the Dirichlet-Neumann iteration in thermal fluid-structure interaction, providing an exact spectral radius formula and confirming its accuracy through numerical experiments.

## Contribution

It derives an exact formula for the spectral radius of the iteration matrix in discretized thermal FSI, linking convergence rate to time step and material properties.

## Key findings

- Convergence rate depends on aspect ratio and material conductivity.
- Decreasing time step improves convergence.
- The 1D formula estimates 2D and nonlinear cases effectively.

## Abstract

We consider the Dirichlet-Neumann iteration for partitioned simulation of thermal fluid-structure interaction, also called conjugate heat transfer. We analyze its convergence rate for two coupled fully discretized 1D linear heat equations with jumps in the material coefficients across these. These are discretized using implicit Euler in time, a finite element method on one domain, a finite volume method on the other one and variable aspect ratio. We provide an exact formula for the spectral radius of the iteration matrix. This shows that for large time steps, the convergence rate is the aspect ratio times the quotient of heat conductivities and that decreasing the time step will improve the convergence rate. Numerical results confirm the analysis and show that the 1D formula is a good estimator in 2D and even for nonlinear thermal FSI applications.