Nested multiscale spaces-based preconditioners for Darcy’s flow in heterogeneous fractured porous media

In this paper, we develop efficient multilevel preconditioners based on nested multiscale subspaces for modeling Darcy’s flow in highly heterogeneous fractured porous media. The proposed framework is applicable to two widely used fracture discretization strategies, namely the discrete fracture model (DFM) and the embedded discrete fracture model (EDFM), and provides robust solvers for the large-scale linear systems arising from both formulations. The core idea is to construct a hierarchy of multiscale subspaces through local spectral decompositions: starting from a coarse partition of the fine grid, a multiscale subspace Q H is constructed by selecting eigenfunctions associated with the smallest eigenvalues of carefully designed local spectral problems, which is then used to build an efficient two-grid preconditioner. To overcome the limitations of two-grid methods for extremely large and strongly heterogeneous fractured-media problems, we further introduce a nested multiscale construction by performing additional spectral decompositions within Q H , naturally leading to a robust three-grid preconditioner. We present theoretical analysis and extensive numerical experiments to assess the robustness and efficiency of the proposed methods. In particular, large-scale three-dimensional fractured porous media problems with up to 160 million degrees of freedom are investigated. The results demonstrate that the proposed multiscale preconditioners achieve high computational efficiency and excellent robustness with respect to fracture complexity and coefficient contrast. Comparisons with algebraic multigrid preconditioners further highlight the advantages of the proposed framework for challenging fractured-media simulations.