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

A multiscale flux basis for mortar mixed discretizations of reduced Darcy-Forchheimer fracture models