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On some oblique derivative problems arising in the fluid flow in porous media. A theoretical and numerical approach

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Résumé

En utilisant une méthode de Baiocchi [1], on ramène l'étude d'une classe de problèmes à frontière libre, traduisant le filtrage d'un liquide à travers un matériau poreux, à l'étude d'un problème non linéaire pour un domaine fixé. On trouve que ce problème n'est pas de type variationel à cause du fait qu'on a une condition aux limites de dérivée oblique avec des discontinuités dans la direction.

On démontre pour les problèmes à frontière libre un résultat d'unicité et en même temps on donne l'existence d'une solution et on introduit un algorithme de type nouveau pour l'approximation de la solution.

Des résultats numériques sont donnés.

Abstract

In this paper we consider some free boundary problems related to the fluid flow in a porous medium. By applying a method due to Baiocchi [1] these problems are reduced to nonlinear problems on a fixed domain. The main difficulty here lies in the fact that such problems are not variational because of jump discontinuities in the direction of the oblique derivative in the boundary condition. We give a uniqueness result and by a constructive method we establish at the same time an existence result and a new algorithm for the numerical solution of the original free boundary problem. Some numerical results are given.

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References

  1. C. Baiocchi, Su un problema di frontiera libera connesso a questioni di idraulica,Ann. Mat. Pura Appl., (IV) XCII (1972), 107–127; preliminary note on C. R. Acad. Sci. Paris,273 (1971), 1215–1217.

    Google Scholar 

  2. C. Baiocchi, Sur quelques problèmes à frontière libre, to appear on Proceedings “Colloque International sur les équations aux dérivées partielles”, Orsay-September 1972.

  3. C. Baiocchi, V. Comincioli, E. Magenes andG. A. Pozzi, Free boundary problems in the theory of fluid flow through porous media: existence and uniqueness theorems,Ann. Mat. Pura Appl., (IV) XCVII (1973), 1–82.

    Google Scholar 

  4. C. Baiocchi, V. Comincioli, L. Guerri andG. Volpi, Free boundary problems in the theory of fluid flow through porous media: a numerical approach, Calcolo, X fasc.1 (1973), 1–86.

    Google Scholar 

  5. C. Baiocchi andE. Magenes, Problemi di frontiera libera in idraulica, to appear inProc. International Congress: Metodi valutativi nella Fisica Matematica, 15–20 dic. 1972, Accad. Naz. Lincei, Roma.

    Google Scholar 

  6. J. Bear, Dynamics of fluids in porous media, American Elsevier, New York, 1972.

    Google Scholar 

  7. V. Benci, Su un problema di filtrazione attraverso un mezzo poroso, to appear inAnn. Mat. pura Appl.

  8. F. E. Browder, Nonlinear maximal monotone operators in Banach spaces,Math. Annalen,175 (1968), 89–113.

    Article  Google Scholar 

  9. J. Cea, Approximation variationelle des problèmes aux limites,Ann. Inst. Fourier,14 (1964), 345–444.

    Google Scholar 

  10. V. Comincioli, L. Guerri andG. Volpi, Analisi numerica di un problema di frontiera libera connesso col moto di un fluido attraverso un mezzo poroso,Publication N.17 of L.A.N. of the C.N.R. Pavia (1971).

  11. V. Comincioli, A theoretical and numerical approach to some free boundary problems, to appear inAnn. Mat. Pura Appl.

  12. M. E. Harr, Groundwater and seepage, McGraw-Hill, New York, 1962.

    Google Scholar 

  13. J. L. Lions andE. Magenes, Non-homogeneous boundary value problems and applications, vol. I, Gundlehren B.181, Springer, Berlin, 1972.

    Google Scholar 

  14. J. C. Miellou, Opérateurs paramonotones, Thèse Grenoble, 1970.

  15. —, Méthodes de Jacobi, Gauss-Seidel, sur (sous)-relaxation par blocs, appliquées à une classe de problèmes non linéaires,C.R. Acad. Sci. Paris,273 (1971), 1257–1260.

    Google Scholar 

  16. J. Neĉas, Sur la coercivité des formes sesquilinéaires elliptiques,Rev. Roumaine de Math. Pure Appl., IX (1964), 47–69.

    Google Scholar 

  17. G. A. Pozzi, On a free-boundary problem arising from fluid flow through a porous medium in the presence of evaporation, to appear inBoll. U.M.I.

  18. R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators,Trans. Amer. Math. Soc.,149 (1970), 75–88.

    Google Scholar 

  19. L. Schwartz, Théorie des distributions, Hermann, Paris, 1966.

    Google Scholar 

  20. L. Tartar, Une nouvelle caracterisation des M matrices, R.I.R.O., 5e année R-3 (1971), 127–128.

  21. A. Torelli, Su un problema di filtrazione da un canale, to appear inRend. Sem. Mat. Padova.

  22. R. S. Varga, Matrix iterative analysis, Prentice Hall, New York, 1962.

    Google Scholar 

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Authors and Affiliations

  1. Istituto di Matematica, Università di Pavia, Pavia, Italia

    Valeriano Comincioli

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  1. Valeriano Comincioli
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Communicated by J. L. Lions

This work was supported by the National Research Council (C.N.R.) of Italy in the frame of the Numerical Analysis Laboratory (L.A.N.) at Pavia.

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Comincioli, V. On some oblique derivative problems arising in the fluid flow in porous media. A theoretical and numerical approach. Appl Math Optim 1, 313–336 (1975). https://doi.org/10.1007/BF01447956

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