We consider an incompressible flow problem in a N-dimensional fractured
porous domain (Darcy’s problem). The fracture is represented by a
(N − 1)-dimensional interface, exchanging fluid with the surrounding
media. In this paper we consider the lowest-order (ℝ T0, ℙ0) Raviart-Thomas mixed finite element
method for the approximation of the coupled Darcy’s flows in the porous media and within
the fracture, with independent meshes for the respective domains. This is achieved thanks
to an enrichment with discontinuous basis functions on triangles crossed by the fracture
and a weak imposition of interface conditions. First, we study the stability and
convergence properties of the resulting numerical scheme in the uncoupled case, when the
known solution of the fracture problem provides an immersed boundary condition. We detail
the implementation issues and discuss the algebraic properties of the associated linear
system. Next, we focus on the coupled problem and propose an iterative porous
domain/fracture domain iterative method to solve for fluid flow in both the porous media
and the fracture and compare the results with those of a traditional monolithic approach.
Numerical results are provided confirming convergence rates and algebraic properties
predicted by the theory. In particular, we discuss preconditioning and equilibration
techniques to make the condition number of the discrete problem independent of the
position of the immersed interface. Finally, two and three dimensional simulations of
Darcy’s flow in different configurations (highly and poorly permeable fracture) are
analyzed and discussed.