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.

On the coupling of 1D and 3D diffusion-reaction equations. Applications to tissue perfusion problems

D'ANGELO, CARLO;QUARTERONI, ALFIO MARIA
2008-01-01

Abstract

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.
2008
coupled one-dimensional; coupled three-dimensional; multiscale modelling; weighted Sobolev spaces; perfusion; one-dimensional fractures; blood flow
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/512943
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