In this paper, we develop a multigrid preconditioner to solve Darcy flow in highly heterogeneous porous media. The key component of the preconditioner is to construct a sequence of nested subspaces $W_{\mathcal{L}}\subset W_{\mathcal{L}-1}\subset\cdots\subset W_1=W_h$. An appropriate spectral problem is defined in the space of $W_{i-1}$, then the eigenfunctions of the spectral problems are utilized to form $W_i$. The preconditioner is applied to solve a positive semidefinite linear system which results from discretizing the Darcy flow equation with the lowest order Raviart-Thomas spaces and adopting a trapezoidal quadrature rule. Theoretical analysis and numerical investigations of this preconditioner will be presented. In particular, we will consider several typical highly heterogeneous permeability fields whose resolutions are up to $1024^3$ and examine the computational performance of the preconditioner in several aspects, such as strong scalability, weak scalability, and robustness against the contrast of the media. We also demonstrate an application of this preconditioner for solving a two-phase flow benchmark problem.

翻译：本文提出了一种多重网格预条件子，用于求解高度非均质多孔介质中的Darcy流。该预条件子的核心在于构造一组嵌套子空间$W_{\mathcal{L}}\subset W_{\mathcal{L}-1}\subset\cdots\subset W_1=W_h$。通过在$W_{i-1}$空间中定义适当的谱问题，并利用其特征函数构建$W_i$。该预条件子应用于求解由最低阶Raviart-Thomas空间离散Darcy流方程并采用梯形求积规则所得的正半定线性系统。本文将给出该预条件子的理论分析与数值研究。特别地，我们将考虑多个分辨率高达$1024^3$的典型强非均质渗透率场，并从强可扩展性、弱可扩展性及对介质对比度的鲁棒性等多个方面检验预条件子的计算性能。此外，我们还展示了该预条件子在求解两相流基准问题中的应用。