Convergence analysis of coupling iterations for the unsteady transmission problem with mixed discretizations

We analyze the convergence rate of the Dirichlet-Neumann iteration for the fully discretized one dimensional unsteady transmission problem. Specifically, we consider the coupling of two linear heat equations on two identical non overlapping intervals. The Laplacian is discretized using finite differences on one interval and finite elements on the other and the implicit Euler method is used for the time discretization. Following previous analysis where finite elements where used on both subdomains, we provide an exact formula for the spectral radius of the iteration matrix for this specific mixed discretizations. We then show that these tend to the ratio of heat conductivities in the semidiscrete spatial limit, but to a factor of the ratio of the products of density and specific heat capacity in the semidiscrete temporal one. In the previous finite element analysis, the same result was obtained in the semidiscrete spatial limit but the factor in the temporal limit was lower. This explains the fast convergence previously observed for cases with strong jumps in the material coefficients. Numerical results confirm the analysis.