In this paper we consider the coupling between two diffusion-reaction problems, one taking place in a three-dimensional domain Ω, the other in a one-dimensional subdomain Λ. This coupled problem is the simplest model of fluid flow in a three-dimensional porous medium featuring fractures that can be described by one-dimensional manifolds. In particular this model can provide the basis for a multiscale analysis of blood flow through tissues, in which the capillary network is represented as a porous matrix, while the major blood vessels are described by thin tubular structures embedded into it: in this case, the model allows the computation of the 3D and 1D blood pressures respectively in the tissue and in the vessels. The mathematical analysis of the problem requires non-standard tools, since the mass conservation condition at the interface between the porous medium and the one-dimensional manifold has to be taken into account by means of a measure term in the 3D equation. In particular, the 3D solution is singular on Λ. In this work, suitable weighted Sobolev spaces are introduced to handle this singularity: the well-posedness of the coupled problem is established in the proposed functional setting. An advantage of such an approach is that it provides a Hilbertian framework which may be used for the convergence analysis of finite element approximation schemes. The investigation of the numerical approximation will be the subject of a forthcoming work.

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Abstract.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 (RT0, 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.Mathematics Subject Classification. 76S05, 35Q86, 65L60.

The aim of this work is to present a reduced mathematical model for describing fluid flow in porous media featuring open channels or fractures. The Darcy’s law is assumed in the porous domain while the Stokes–Brinkman equations are considered in the fractures. We address the case of fractures whose thickness is very small compared to the characteristic diameter of the computational domain, and describe the fracture as if it were an interface between porous regions. We derive the corresponding interface model governing the fluid flow in the fracture and in the porous media, and establish the well-posedness of the coupled problem. Further, we introduce a finite element scheme for the approximation of the coupled problem, and discuss solution strategies. We conclude by showing the numerical results related to several test cases and compare the accuracy of the reduced model compared with the non-reduced one

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