In this paper, we present a monolithic multigrid method for the efficient solution of flow problems in fractured porous media. Specifically, we consider a mixed-dimensional model which couples Darcy flow in the porous matrix with Forchheimer flow within the fractures. A suitable finite volume discretization permits to reduce the coupled problem to a system of nonlinear equations with a saddle point structure. In order to solve this system, we propose a full approximation scheme (FAS) multigrid solver that appropriately deals with the mixed-dimensional nature of the problem by using mixed-dimensional smoothing and inter-grid transfer operators. Numerical experiments show that the proposed multigrid method is robust with respect to the fracture permeability, the Forchheimer coefficient and the mesh size. The case of several possibly intersecting fractures in a heterogeneous porous medium is also discussed.

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Computers and Mathematics with Applications
Efficient numerical methods for deformable porous media. Application to carbon dioxide storage
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Arraras, A., Gaspar, F., Portero, L., & Rodrigo, C. (2019). Geometric multigrid methods for Darcy–Forchheimer flow in fractured porous media. Computers and Mathematics with Applications, 78(9), 3139–3151. doi:10.1016/j.camwa.2019.04.031