In this article, a two-level overlapping domain decomposition preconditioner is developed for solving linear algebraic systems obtained from simulating Darcy flow in high-contrast media. Our preconditioner starts at a mixed finite element method for discretizing the partial differential equation by Darcy's law with the no-flux boundary condition and is then followed by a velocity elimination technique to yield a linear algebraic system with only unknowns of pressure. Then, our main objective is to design a robust and efficient domain decomposition preconditioner for this system, which is accomplished by engineering a multiscale coarse space that is capable of characterizing high-contrast features of the permeability field. A generalized eigenvalue problem is solved in each non-overlapping coarse element in a communication-free manner to form the global solver, which is accompanied by local solvers originated from additive Schwarz methods but with a non-Galerkin discretization to derive the two-level preconditioner. We provide a rigorous analysis that indicates that the condition number of the preconditioned system could be bounded above with several assumptions. Extensive numerical experiments with various types of three-dimensional high-contrast models are exhibited. In particular, we study the robustness against the contrast of the media as well as the influences of numbers of eigenfunctions, oversampling sizes, and subdomain partitions on the efficiency of the proposed preconditioner. Besides, strong and weak scalability performances are also examined.

翻译：本文针对高对比度介质中模拟达西流所生成的线性代数系统，提出了一种两重重叠区域分解预条件子。该预条件子首先采用混合有限元方法对带无通量边界条件的达西定律偏微分方程进行离散，随后通过速度消去技术得到仅含压力未知量的线性代数系统。我们的主要目标是为该系统设计一种稳健高效的区域分解预条件子，通过构造能表征渗透率场高对比度特征的多尺度粗空间来实现。在每个非重叠粗单元中，以无通信方式求解广义特征值问题以形成全局求解器，并辅以基于加性Schwarz方法（采用非Galerkin离散）的局部求解器，从而构建两重预条件子。严格的理论分析表明，在若干假设条件下，预条件系统的条件数存在上界。本文通过多种三维高对比度模型进行了大量数值实验，重点研究了预条件子对介质对比度的鲁棒性，以及特征函数数量、过采样尺寸和子区域划分对预条件子效率的影响。此外，还考察了其强可扩展性与弱可扩展性表现。