We consider a numerical solution of the mixed dimensional discrete fracture model with highly conductive fractures. We construct an unstructured mesh that resolves lower dimensional fractures on the grid level and use the finite element approximation to construct a discrete system with an implicit time approximation. Constructing an efficient preconditioner for the iterative method is challenging due to the high resolution of the process and high-contrast properties of fractured porous media. We propose a two-grid algorithm to construct an efficient solver for mixed-dimensional problems arising in fractured porous media and use it as a preconditioner for the conjugate gradient method. We use a local pointwise smoother on the fine grid and carefully design an adaptive multiscale space for coarse grid approximation based on a generalized eigenvalue problem. The construction of the basis functions is based on the Generalized Multiscale Finite Element Method, where we solve local spectral problems with adaptive threshold to automatically identify the dominant modes which correspond to the very small eigenvalues. We remark that such spatial features are automatically captured through our local spectral problems, and connect these to fracture information in the global formulation of the problem. Numerical results are given for two fracture distributions with 30 and 160 fractures, demonstrating iterative convergence independent of the contrast of fracture and porous matrix permeability.

翻译：我们考虑高导流裂隙混合维离散裂隙模型的数值求解。我们构建了一个在网格层面解析低维裂隙的非结构化网格，并采用有限元近似结合隐式时间离散构建离散系统。由于模拟过程的高分辨率以及裂隙多孔介质的高对比度特性，为迭代方法构建高效预条件子具有挑战性。本文提出一种两重网格算法，用于构建裂隙多孔介质中混合维问题的高效求解器，并将其作为共轭梯度法的预条件子。我们在细网格上采用局部点态光滑子，并基于广义特征值问题精心设计了一个用于粗网格近似的自适应多尺度空间。基函数的构建基于广义多尺度有限元方法，通过求解具有自适应阈值的局部谱问题来自动识别对应于极小特征值的主导模态。需要指出的是，此类空间特征通过我们的局部谱问题被自动捕获，并在问题的全局表述中与裂隙信息建立关联。数值实验针对包含30条和160条裂隙的两种裂隙分布进行，结果表明迭代收敛性不依赖于裂隙与多孔基质渗透率的对比度。