Our research focuses on the development of domain decomposition preconditioners tailored for second-order elliptic partial differential equations. Our approach addresses two major challenges simultaneously: i) effectively handling coefficients with high-contrast and multiscale properties, and ii) accommodating irregular domains in the original problem, the coarse mesh, and the subdomain partition. The robustness of our preconditioners is crucial for real-world applications, such as the efficient and accurate modeling of subsurface flow in porous media and other important domains. The core of our method lies in the construction of a suitable partition of unity functions and coarse spaces utilizing local spectral information. Leveraging these components, we implement a two-level additive Schwarz preconditioner. We demonstrate that the condition number of the preconditioned systems is bounded with a bound that is independent of the contrast. Our claims are further substantiated through selected numerical experiments, which confirm the robustness of our preconditioners.

翻译：我们的研究聚焦于针对二阶椭圆型偏微分方程的区域分解预条件子开发。该方法同时应对两大挑战：一）有效处理具有高对比度和多尺度特性的系数，二）适应原始问题、粗网格及子区域划分中的不规则域。预条件子的鲁棒性对实际应用至关重要，例如多孔介质中地下流的高效精确模拟及其他重要领域。方法核心在于利用局部谱信息构建适当的分区单位函数和粗空间。基于这些组件，我们实现了一种两级加性Schwarz预条件子。我们证明了预条件系统的条件数具有与对比度无关的有界性。通过精选数值实验进一步验证了预条件子的鲁棒性。