Dirichlet-Neumann waveform relaxation for heterogeneous heat equations: continuous and time-discrete $L^2$ analysis

Abstract

We consider two coupled linear heat equations on different spatial domains that interact through a lower dimensional interface. This models conjugate heat transfer. The problem is solved using Dirichlet-Neumann waveform relaxation. This allows us to couple separate codes for the subproblems, a so-called partitioned approach. Our overall goal is to develop more efficient partitioned methods, and to this end, we want reliable error estimates. We use an exponentially weighted Fourier technique to derive new error estimates in $L^2$ for finite time $T$ in both continuous and time-discrete settings. We identify an optimized relaxation parameter that guarantees superlinear convergence. Our new continuous estimate predicts linear convergence when $k^2/T$ is small, and superlinear when $k^2/T$ is large, with $k$ the iteration index. Our new time-discrete estimate closely mirrors its continuous counterpart for small dt. Otherwise, we obtain linear convergence for large $k$. We also show that convergence is fast when the contrast is large, provided that the small physical parameter domain (e.g. air) is using the Dirichlet transmission condition, and the large physical parameter domain (e.g. steel) is using the Neumann transmission condition in the Dirichlet-Neumann waveform relaxation method. Numerical experiments confirm these findings,and we demonstrate the relevance on the analysis for a nonlinear conjugate heat transfer test case.