An adaptive two-grid preconditioner for flow in fractured porous media

Abstract: 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.