AI 中文总结
针对高对比度多尺度介质的时变问题，提出基于NLMC方法的两层预条件子，通过分区域分量构造粗空间，在保证鲁棒性的同时降低计算成本，经数值实验验证其性能优异。
AI 中文摘要
在本研究中，我们针对高对比度多尺度介质中的时变问题，提出了一种两层重叠预条件子。我们给出了基于多尺度方法的粗空间构造，重点关注松弛非local多连续介质（NLMC）方法。NLMC空间可分为代表高渗透率区域和低渗透率背景的分量。我们证明，完整的NLMC空间能有效预处理异刚度算子，而仅高渗透率分量即可捕获依赖于对比度的全局模式。对于合适的小时步尺寸，质量矩阵控制低渗透率贡献，两层预条件子的全局粗校正仅需NLMC空间中的高渗透率分量；对于一般时步尺寸，使用完整NLMC空间或用标准多尺度空间扩充高渗透率分量可维持性能。所提出的粗空间构造在保留对系数对比度和细尺度分辨率的鲁棒性的同时降低了计算成本。为进一步提升效率，多尺度基函数可通过迭代求解松弛能量极小化公式来构造。我们通过多个数值实验验证了该方法的鲁棒性、效率和可扩展性。
英文摘要
In this work, we develop a two-level overlapping preconditioner for time-dependent problems in high-contrast multiscale media. We present a coarse-space construction based on multiscale methods, with emphasis on the relaxed nonlocal multicontinuum (NLMC) method. The NLMC space can be separated into components representing the high-permeability regions and the low-permeability background. We show that the complete NLMC space effectively preconditions the heterogeneous stiffness operator, whereas the high-permeability component alone captures the contrast-dependent global modes. For suitable small time-step sizes, the mass matrix controls the low-permeability contribution and only the high-permeability component in NLMC space is required for the global coarse correction in the two-level preconditioner. For general time-step sizes, performance can be maintained by using the full NLMC space or augmenting the high-permeability component with a standard multiscale space. The proposed coarse-space construction lowers the computational cost while preserving robustness with respect to coefficient contrast and fine-scale resolution. To further improve efficiency, the multiscale basis functions can be constructed by iteratively solving the relaxed energy-minimizing formulation. We demonstrate the robustness, efficiency and scalability of the proposed method through several numerical experiments.
