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 the interface. The heat equations are discretized using an implicit Euler scheme in time, whereas a finite element method on one domain and a finite volume method with variable aspect ratio on the other one are used in space. We provide an exact formula for the spectral radius of the iteration matrix. The formula indicates 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 very good estimator in 2D and even for nonlinear thermal FSI applications.