The Schwarz domain decomposition (SDD) method is known for its high efficacy in solving large-scale systems of partial differential equations, primarily due to its parallelizability. However, the method's reliance on iteration introduces substantial computational expenses. In this study, we propose a reduced-order Schwarz domain decomposition (ROSDD) method tailored specifically for the convection-diffusion equation. Utilizing the proper orthogonal decomposition (POD) technique, the ROSDD method significantly reduces computational costs by considering only a small number of unknowns. Since the equation typically exhibits convection-dominated behavior, we employ the characteristic finite element discretization scheme. Consequently, we derive optimal a priori error estimates to assess the accuracy of our approach. Finally, we conduct some numerical experiments to validate the superiority of the ROSDD method.

中文翻译：

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基于POD的对流扩散方程降阶Schwarz域分解方法

Schwarz 域分解 (SDD) 方法因其在求解大规模偏微分方程组方面的高效性而闻名，这主要是由于其可并行性。然而，该方法对迭代的依赖会带来大量的计算开销。在这项研究中，我们提出了一种专门针对对流扩散方程定制的降阶施瓦茨域分解（ROSDD）方法。ROSDD 方法利用适当的正交分解 (POD) 技术，仅考虑少量未知数，从而显着降低计算成本。由于该方程通常表现出对流主导的行为，因此我们采用特征有限元离散化方案。因此，我们得出最佳的先验误差估计来评估我们方法的准确性。最后，我们进行了一些数值实验来验证ROSDD方法的优越性。