Geometric multigrid methods for Darcy-Forchheimer flow in fractured porous media
It provides an efficient solver for a challenging mixed-dimensional flow problem in fractured porous media, which is important for applications like geothermal energy and oil recovery.
The paper presents a monolithic multigrid method for solving coupled Darcy-Forchheimer flow in fractured porous media, demonstrating robustness with respect to fracture permeability, Forchheimer coefficient, and mesh size through numerical experiments.
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. Remarkably, the nonlinearity is localized in the fractures, and no coupling between the porous matrix and the fracture unknowns is needed in the smoothing procedure. Numerical experiments show that the proposed multigrid method is robust with respect to the fracture permeability, the Forchheimer coefficient and the mesh size.