Dirichlet-Neumann waveform relaxation for heterogeneous heat equations: continuous and time discrete L2 analysis

Problem Statement
The paper addresses the numerical solution of coupled linear heat equations defined on distinct spatial domains interacting through a lower-dimensional interface. This setup models conjugate heat transfer problems, such as those found in multiphysics applications like fluid-structure interaction or heat transfer in composite materials. The authors focus on solving these problems using a partitioned approach, specifically the Dirichlet–Neumann Waveform Relaxation (DNWR) method. In this framework, separate solvers are used for each subdomain, communicating only through interface conditions (Dirichlet on one side, Neumann on the other). The primary challenge addressed is the development of reliable error estimates to ensure the efficiency and stability of these coupling algorithms, particularly regarding the convergence behavior dependent on relaxation parameters, physical material properties, and the length of the time window.

Methodology
The authors employ an exponentially weighted Fourier transform technique, originally proposed in [26], to analyze the convergence of the DNWR method. This approach is applied in two distinct settings:

1. Continuous Setting: The analysis is performed on the continuous time domain for a finite time interval $T$. The method involves transforming the error equations associated with the DNWR iteration into the frequency domain using an exponentially weighted Fourier transform with a weight parameter $r > 0$.
2. Time Discrete Setting: The analysis is extended to the time-discrete case using the implicit Euler scheme. Here, an exponentially weighted discrete-time Fourier transform is utilized, incorporating the time step size $\Delta t$.

In both settings, the authors derive $L^2$ error estimates for arbitrary relaxation parameters $\theta$. They subsequently identify an optimized relaxation parameter, $\theta^*$, which depends on the physical parameters (volumetric heat capacity $\alpha$ and thermal conductivity $\lambda$) and domain sizes. The convergence analysis involves solving a min-max problem to bound the contraction factor, distinguishing between cases where the physical parameters satisfy specific scaling relations (e.g., $\sqrt{\alpha_1/\lambda_1}a > \sqrt{\alpha_2/\lambda_2}b$).

Key Contributions

• New Error Estimates: The paper derives new $L^2$ error estimates for both continuous and time-discrete DNWR methods applied to heterogeneous heat equations on finite time windows. These estimates are valid for arbitrary relaxation parameters.
• Optimized Relaxation Parameter: An optimized relaxation parameter $\theta^*$ is identified that guarantees superlinear convergence for small time windows $T$. This parameter is shown to be independent of the mesh size for high frequencies.
• Superlinear vs. Linear Convergence: The analysis reveals that convergence behavior is time-window dependent. For large $T$, the method exhibits linear convergence, whereas for small $T$, it exhibits superlinear convergence.
• Time Discrete vs. Continuous Discrepancy: A critical contribution is the identification of the relationship between continuous and discrete convergence. The authors show that while the continuous estimate predicts superlinear convergence for small $T$, the time-discrete estimate only preserves this superlinear behavior if the time step $\Delta t$ is sufficiently small relative to $T$. If $\Delta t$ is too large, the discrete method converges linearly, even when the continuous theory suggests superlinear convergence.
• Physical Parameter Influence: The study demonstrates that convergence is significantly faster when the domain with smaller physical parameters (e.g., air) is assigned the Dirichlet transmission condition, and the domain with larger physical parameters (e.g., steel) is assigned the Neumann condition.

Results

• Theoretical Bounds: Theorems 2 and 3 provide explicit error bounds for the continuous case, showing a linear convergence factor multiplied by an exponential term that decays superlinearly for small $T$. Theorems 5 and 6 provide analogous bounds for the time-discrete case, introducing a dependency on $\Delta t$.
• Numerical Validation: Numerical experiments in both 1D and 2D confirm the theoretical findings.
  • In low-contrast cases, the time-discrete estimates approach the continuous estimates as $\Delta t \to 0$.
  • In high-contrast cases (air-steel vs. steel-air), the "air-steel" configuration (Dirichlet on air, Neumann on steel) converges rapidly (reaching errors of $10^{-12}$ in fewer than 3 iterations), while the reverse configuration is significantly slower.
  • The numerical results validate that for small $T$, a sufficiently small $\Delta t$ is required to observe the superlinear convergence predicted by the continuous analysis.
  • The 2D experiments suggest that the 1D theoretical estimates remain effective predictors for higher-dimensional problems, at least for the initial iterations.

Significance
The paper claims to close existing gaps in the literature by providing continuous and time-discrete error estimates that explicitly depend on the time window length $T$ and the time step $\Delta t$. It highlights that relying solely on continuous analysis can be misleading for time-discrete implementations, particularly when the simulation is split into small time windows. The findings provide practical guidance for selecting relaxation parameters and time step sizes to ensure efficient convergence in partitioned multiphysics simulations. The identification of the optimal assignment of transmission conditions based on physical parameter contrasts offers a concrete strategy for accelerating convergence in heterogeneous heat transfer problems.