This paper explored the optimized Schwarz method (OSM) for solving contact conductance heat transfer problems, with a particular emphasis on the impact of thermal contact resistance (TCR) on the convergence behavior of the OSM algorithm when the standard Robin transmission condition is applied. The presence of TCR introduces a challenging optimization problem, which was rigorously addressed to determine the optimal Robin parameter and describe the corresponding convergence behavior. Our analysis yielded several novel findings. First, an increase in TCR results in faster convergence of the OSM algorithm. Second, mesh-independent convergence was achieved in an asymptotic sense, contrasting with the mesh-dependent convergence observed in the absence of TCR. Third, unlike the deceleration caused by strong heterogeneity in the TCR-free scenario, increased heterogeneity contrast accelerates convergence. Thermal conductivity also contributes to convergence enhancement in a manner analogous to the effect of heterogeneity. These theoretical results were validated through numerical experiments, demonstrating the significant influence of TCR on the performance of the OSM algorithm in contact conductance heat transfer problems.

The non-Fourier heat transfer in heterogeneous media is crucial for material science and biomedical engineering. The optimized Schwarz waveform relaxation (OSWR) method is an efficient approach for solving such problems due to its divide-and-conquer strategy. Despite the wave-type nature of non-Fourier heat transfer, the short phase-lag time leads to more parabolic-like behavior. To address this, in the OSWR method, we employed Robin boundary conditions to transmit information along the interface. Using Fourier analysis, we derived and rigorously optimized the convergence factors of the OSWR algorithm with scaled Robin and Robin-Robin transmission conditions. The resulting optimized transmission parameters were provided in explicit form for direct application in the OSWR algorithm, along with corresponding convergence factor estimates. Interestingly, the results show that a larger heterogeneity contrast actually accelerates the convergence, rather than deteriorating it. Furthermore, the OSWR algorithm with the Robin-Robin condition exhibits mesh-independent convergence asymptotically. However, the presence of the phase-lag time is found to slow down the convergence of the OSWR algorithm. These theoretical findings were validated through numerical experiments.