  In this work, we consider compressible single-phase flow problems in a porous
media containing a fracture. In the latter, a non-linear pressure-velocity
relation is prescribed. Using a non-overlapping domain decomposition procedure,
we reformulate the global problem into a non-linear interface problem. We then
introduce two new algorithms that are able to efficiently handle the
non-linearity 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 the non-linearity, 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 pre-computing, in an offline
phase, a multiscale flux basis (the linear Robin-to-Neumann co-dimensional
map), that represent the flux exchange between the fracture and the matrix. We
present extensive theoretical findings and in particular, the stability and the
convergence of both schemes are obtained, where user given parameters are
optimized to minimise the number of iterations. Examples on two important
fracture models are computed with the library PorePy and agree with the
developed theory.
