@article {27772,
	title = {Robust linear domain decomposition schemes for reduced non-linear fracture flow models},
	journal = {SIAM J. Numer. Anal.},
	volume = {59},
	number = {1},
	year = {2021},
	pages = {583-612},
	publisher = {SIAM},
	abstract = {In this work, we consider compressible single-phase flow problems in a porous medium containing a fracture. In the fracture, a nonlinear pressure-velocity relation is prescribed. Using a non-overlapping domain decomposition procedure, we reformulate the global problem into a nonlinear interface problem. We then introduce two new algorithms that are able to efficiently handle the nonlinearity and the coupling between the fracture and the matrix, both based on linearization by the so-called L-scheme. The first algorithm, named MoLDD, uses the L-scheme to resolve for the nonlinearity, requiring at each iteration to solve the dimensional coupling via a domain decomposition approach. The second algorithm, called ItLDD, uses a sequential approach in which the dimensional coupling is part of the linearization iterations. For both algorithms, the computations are reduced only to the fracture by precomputing, in an offline phase, a multiscale flux basis (the linear Robin-to-Neumann codimensional map), that represent the flux exchange between the fracture and the matrix. We present extensive theoretical findings X and in particular, t. The stability and the convergence of both schemes are obtained, where user-given parameters are optimized to minimize the number of iterations. Examples on two important fracture models are computed with the library PorePy and agree with the developed theory. },
	doi = {10.1137/19M1268392},
	url = {https://doi.org/10.1137/19M1268392},
	author = {Ahmed, Elyes and Fumagalli, Alessio and Budisa, Ana and Keilegavlen, Eirik and Jan Martin {Nordbotten} and Florin Adrian {Radu}}
}
@article {27771,
	title = {A multiscale flux basis for mortar mixed discretizations of reduced Darcy{\textendash}Forchheimer fracture models},
	journal = {Computer Methods in Applied Mechanics and Engineering},
	volume = {354},
	year = {2019},
	month = {05/2019},
	pages = {16 - 36},
	publisher = {Elsevier},
	abstract = {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{\textendash}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{\textendash}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 (DOF) 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.},
	issn = {00457825},
	doi = {10.1016/j.cma.2019.05.034},
	url = {https://www.sciencedirect.com/science/article/pii/S0045782519303044},
	author = {Ahmed, Elyes and Fumagalli, Alessio and Budisa, Ana}
}
