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ON THE COUPLING OF 1D AND 3D DIFFUSION-REACTION EQUATIONS: APPLICATION TO TISSUE PERFUSION PROBLEMS

  • D'ANGELO, C. MOX, Department of Mathematics "F. Brioschi", Politecnico di Milano, Piazza Leonardo da Vinci 32 — 20133 Milano, Italy
  • QUARTERONI, A. CMCS, École Politechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
  • 2011-11-21
Published in:
  • Mathematical Models and Methods in Applied Sciences. - World Scientific Pub Co Pte Lt. - 2008, vol. 18, no. 08, p. 1481-1504
Modelling and Simulation
Applied Mathematics
English 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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  • English
Open access status
green
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https://sonar.rero.ch/global/documents/194014
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