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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., Comment: 29 pages, 20 figures

An important challenge when coupling two different time dependent problems is to increase parallelization in time. We suggest a multirate Neumann-Neumann waveform relaxation algorithm to solve two heterogeneous coupled heat equations. In order to fix the mismatch produced by the multirate feature at the space-time interface a linear interpolation is constructed. The heat equations are discretized using a finite element method in space, whereas two alternative time integration methods are used: implicit Euler and SDIRK2. We perform a one-dimensional convergence analysis for the nonmultirate fully discretized heat equations using implicit Euler to find the optimal relaxation parameter in terms of the material coefficients, the stepsize and the mesh resolution. This gives a very efficient method which needs only two iterations. Numerical results confirm the analysis and show that the 1D nonmultirate optimal relaxation parameter is a very good estimator for the multirate 1D case and even for multirate and nonmultirate 2D examples using both implicit Euler and SDIRK2., Comment: 32 pages, 12 figures

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. (Less)

We analyze the convergence rate of the Dirichlet-Neumann iteration for the fully discretized unsteady transmission problem. Specifically, we consider the coupling of two linear heat equations on two identical non overlapping domains with jumps in the material coefficients across these. In this context, we derive the iteration matrix of the coupled problem. In the 1D case, the spectral radius of the iteration matrix tends to the ratio of heat conductivities in the semidiscrete spatial limit, but to the ratio of the products of density and specific heat capacity in the semidiscrete temporal one. This explains the fast convergence previously observed for cases with strong jumps in the material coefficients.