Dirichlet--Neumann waveform relaxation for heterogeneous heat equations: fully 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 the subproblems to be solved using separate codes, a so called partitioned approach. Our overall goal is to develop more efficient partitioned methods, and to this end, we want reliable error estimates. Here, we use an exponentially weighted Fourier technique to derive new error estimates in $l^2$ for finite time $T$ in the fully discrete setting. These describe both linear and superlinear behavior. We show that the fully discrete estimate is close to a previously obtained time discrete estimate and independent of $\Delta x_1$ and $\Delta x_2$ when the CFL number in the subsolvers is large. We also show that the convergence behaviour depends on the ratio $\Delta x_1/\Delta x_2$ when the CFL number is small. Our numerical experiments show that the fully discrete estimate is accurate across a wide range of grid sizes $\Delta x_1,\Delta x_2$, time step sizes $\Delta t$ and $T$.