Geometric multigrid methods for Darcy–Forchheimer flow in fractured porous media

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.Francisco J. Gaspar has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 705402, POROSOS. The work of Carmen Rodrigo is supported in part by the FEDER / MINECO, Spain project MTM2016-75139-R. The work of Andrés Arrarás and Laura Portero is supported in part by the FEDER / MINECO, Spain projects MTM2014-52859-P and MTM2016-75139-R