Iterative Substructuring Methods for Advection — Diffusion Problems in Heterogeneous Media

  • Paolo Zunino
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 35)


This work is devoted to the numerical approximation of a system of advection-diffusion equations set in adjacent domains and coupled with nonstandard matching conditions. The specific application of the model at hand is the study of the transfer of chemicals through media of heterogeneous nature, for instance a free fluid and a porous medium.

After a brief description of the model, we focus our attention on its numerical treatment. In particular, since our model couples subproblems in different media, we study an iterative procedure where the solutions provided separately on each sub-domain are suitably matched (a so called iterative substructuring method). More precisely, we consider a strategy based on Robin interface conditions for both subdomains. The convergence of this iterative strategy is analyzed at both the continuous and the discrete level. Moreover, an algebraic reinterpretation of this technique is provided, leading to the definition of optimal preconditioners for the linear system arising from the discretization of the global problem. Finally, numerical results are presented, in order to assess the computational efficiency of the numerical methods proposed.


Iterative Method Diffusion Problem Free Fluid Domain Decomposition Method Trace Inequality 
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  1. 1.
    Yves Achdou, Patrick Le Tallec,FrédéricNataf, and Marina Vidrascu, A domain decomposition preconditioner for an advection-diffusion problem Comput. Methods Appl. Mech. Engrg. 184 (2000), no. 2–4, 2–4.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. Alonso, R.T. Trotta, and A. Valli, Coercive domain decomposition algorithms for advection-diffusion equations and systems J. Comput. Appl. Math. 96 (1998), 51–76.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M. Discacciati, E. Miglio, and A. Quarteroni, Mathematical and numerical models for coupling surface and groundwater flows Applied Numerical Mathematics 43 (2002), 57–74.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    G. Karner, K. Perktold, and H.P. Zehentner, Infra and Extracorporeal Cardiovascular Fluid Dynamics, ch. 7: Mass transport in large arteries and through the arterial wall, pp. 209–247, P. Verdonck and K. Perktold eds., WIT-Press - Computational Mechanics Publications, 2000.Google Scholar
  5. 5.
    J. L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications vol. 1,2,3, Dunod, Paris, 1968.Google Scholar
  6. 6.
    A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics Springer, New York, 2000.Google Scholar
  7. 7.
    A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press 1999.zbMATHGoogle Scholar
  8. 8.
    A. Quarteroni, A. Veneziani, and P Zunino, A domain decomposition method for advection-diffusion processes with application to blood solutes SIAM J. Sci. Comput. 23 (2002), no. 6, 1959–1980.MathSciNetzbMATHGoogle Scholar
  9. 9.
    A. Quarteroni, A. Veneziani, and P Zunino, AMathematical and numerical modelling of solute dynamics in blood flow and arterial wallsSIAM J. Numer. Anal. 39 (2002), no. 5, 1488–1511.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing Company, 1996.Google Scholar
  11. 11.
    P. Zunino, Mathematical and numerical modeling of mass transfer in the vascular systemPh.D. thesis, Ecole Polythéchnique Fédérale de Lausanne, 2002.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Paolo Zunino
    • 1
  1. 1.Modeling and Scientific Computing Laboratory (MOCS)Institute of Mathematics, Ecole Polytéchnique Fédérale de Lausanne (EPFL)Switzerland

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