  In this paper, a multiscale flux basis algorithm is developed to efficiently
solve a flow problem in fractured porous media. Here, we take into account a
mixed-dimensional setting of the discrete fracture matrix model, where the
fracture network is represented as lower-dimensional object. We assume the
linear Darcy model in the rock matrix and the non-linear Forchheimer model in
the fractures. In our formulation, we are able to reformulate the
matrix-fracture problem to only the fracture network problem and, therefore,
significantly reduce the computational cost. The resulting problem is then a
non-linear interface problem that can be solved using a fixed-point or
Newton-Krylov methods, which in each iteration require several solves of Robin
problems in the surrounding rock matrices. To achieve this, the flux exchange
(a linear Robin-to-Neumann co-dimensional mapping) between the porous medium
and the fracture network is done offline by pre-computing a multiscale flux
basis that consists of the flux response from each degree of freedom on the
fracture network. This delivers a conserve for the basis that handles the
solutions in the rock matrices for each degree of freedom in the fractures
pressure space. Then, any Robin sub-domain problems are replaced by linear
combinations of the multiscale flux basis during the interface iteration. The
proposed approach is, thus, agnostic to the physical model in the fracture
network. Numerical experiments demonstrate the computational gains of
pre-computing the flux exchange between the porous medium and the fracture
network against standard non-linear domain decomposition approaches.
