Heterogeneous Optimized Schwarz Methods for Heat Conduction in Composites with Thermal Contact Resistance

Heat transfer in composites is crucial in engineering, where imperfect layer contact induces thermal contact resistance (TCR), causing interfacial temperature jumps. We propose to solve this numerically using the optimized Schwarz method (OSM), which decouples the heterogeneous problem into homogeneous subproblems. This approach avoids ill-conditioned systems typical of monolithic methods under high contrast and interface discontinuities. Convergence of the algorithm with a standard Robin transmission condition is established via energy estimates and Fourier analysis. For accelerated convergence, a scaled Robin condition is introduced, with rigorous optimization of its free parameter. Key findings emerge due to TCR: first, larger TCR values speed up OSM convergence, achieving asymptotic mesh-independence. This contrasts with the mesh-dependent behavior observed without TCR. Second, greater heterogeneity contrast enhances convergence; third, unlike TCR-free cases, higher thermal conductivity also promotes convergence, similar to heterogeneity; finally, the scaled Robin condition outperforms the standard one in both theory and practice. Numerical tests validate the results and demonstrate the method's potential for nonlinear problems on irregular domains.