In this paper, a new waveform relaxation variant of the Dirichlet–Neumann algorithm is introduced for general parabolic problems as well as for the second-order wave equation for decompositions with multiple subdomains. The method is based on a non-overlapping decomposition of the domain in space, and the iteration involves subdomain solves in space-time with transmission conditions of Dirichlet and Neumann type to exchange information between neighboring subdomains. Regarding the convergence of the algorithm, two main results are obtained when the time window is finite: for the heat equation, the method converges superlinearly, whereas for the wave equation, it converges after a finite number of iterations. The analysis is based on Fourier–Laplace transforms and detailed kernel estimates, which reveals the precise dependence of the convergence on the size of the subdomains and the time window length. Numerical experiments are presented to illustrate the performance of the algorithm and to compare its convergence behaviour with classical and optimized Schwarz Waveform Relaxation methods. Experiments involving heterogeneous coefficients and non-matching time grids, which are not covered by the theory, are also presented.

中文翻译：

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多子域抛物线和双曲线问题的 Dirichlet-Neumann 波形松弛方法

在本文中，Dirichlet-Neumann 算法的新波形松弛变体被引入用于一般抛物线问题以及用于具有多个子域的分解的二阶波动方程。该方法基于空间域的非重叠分解，迭代涉及时空子域求解，具有狄利克雷和诺依曼类型的传输条件，以在相邻子域之间交换信息。关于算法的收敛性，当时间窗有限时，得到两个主要结果：对于热方程，该方法超线性收敛，而对于波动方程，经过有限次迭代后收敛。该分析基于傅立叶-拉普拉斯变换和详细的核估计，这揭示了收敛对子域大小和时间窗口长度的精确依赖性。提供了数值实验来说明算法的性能，并将其收敛行为与经典和优化的 Schwarz 波形松弛方法进行比较。还介绍了理论未涵盖的涉及异构系数和非匹配时间网格的实验。