Diffusion and propagation problems in some ramified domains with a fractal boundary

Achdou, Yves; Sabot, Christophe; Tchou, Nicoletta

ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 4, pp. 623-652.

This paper is devoted to some elliptic boundary value problems in a self-similar ramified domain of ${ℝ}^2$ with a fractal boundary. Both the Laplace and Helmholtz equations are studied. A generalized Neumann boundary condition is imposed on the fractal boundary. Sobolev spaces on this domain are studied. In particular, extension and trace results are obtained. These results enable the investigation of the variational formulation of the above mentioned boundary value problems. Next, for homogeneous Neumann conditions, the emphasis is placed on transparent boundary conditions, which allow the computation of the solutions in the subdomains obtained by stopping the geometric construction after a finite number of steps. The proposed methods and algorithms will be used numerically in forecoming papers.

Classification : 28A80, 35J05, 35J25, 65N
Mots clés : domains with fractal boundaries, Helmholtz equation, Neumann boundary conditions, transparent boundary conditions

[1] Y. Achdou, C. Sabot and N. Tchou, A multiscale numerical method for Poisson problems in some ramified domains with a fractal boundary. SIAM Multiscale Model. Simul. (2006) (accepted for publication). | MR

[2] Y. Achdou, C. Sabot and N. Tchou, Transparent boundary conditions for Helmholtz equation in some ramified domains with a fractal boundary. J. Comput. Phys. (2006) (in press). | MR | Zbl

[3] R.A. Adams, Sobolev spaces. Academic Press, New York-London (1975). Pure Appl. Math. 65. | MR | Zbl

[4] H. Brezis, Analyse fonctionnelle. Collection Mathématiques Appliquées pour la Maîtrise. Théorie et applications. Masson, Paris, 1983. | MR | Zbl

[5] M. Felici, Physique du transport diffusif de l'oxygène dans le poumon humain. Ph.D. thesis, École Polytechnique (2003).

[6] M. Gibbons, A. Raj and R.S. Strichartz, The finite element method on the Sierpinski gasket. Constr. Approx. 17 (2001) 561-588. | Zbl

[7] P. Grisvard, Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics 24, Pitman (Advanced Publishing Program), Boston, MA (1985). | MR | Zbl

[8] J.E. Hutchinson, Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981) 713-747. | Zbl

[9] P.W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147 (1981) 71-88. | Zbl

[10] A. Jonsson and H. Wallin, Function spaces on subsets of ${𝐑}^n$. Math. Rep. 2 (1984) xiv+221. | MR | Zbl

[11] J.B. Keller and D. Givoli, Exact nonreflecting boundary conditions. J. Comput. Phys. 82 (1989) 172-192. | Zbl

[12] M.R. Lancia, A transmission problem with a fractal interface. Z. Anal. Anwendungen 21 (2002) 113-133. | Zbl

[13] M.R. Lancia, Second order transmission problems across a fractal surface. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 27 (2003) 191-213.

[14] B.B. Mandelbrodt, The fractal geometry of nature. Freeman and Co (1982). | MR | Zbl

[15] B. Mauroy, M. Filoche, J.S. Andrade and B. Sapoval, Interplay between flow distribution and geometry in an airway tree. Phys. Rev. Lett. 90 (2003).

[16] B. Mauroy, M. Filoche, E.R. Weibel and B. Sapoval, The optimal bronchial tree is dangerous. Nature 427 (2004) 633-636.

[17] V.G. Maz'Ja, Sobolev spaces. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin (1985). Translated from the Russian by T.O. Shaposhnikova. | Zbl

[18] U. Mosco, Energy functionals on certain fractal structures. J. Convex Anal. 9 (2002) 581-600. | Zbl

[19] U. Mosco and M.A. Vivaldi, Variational problems with fractal layers. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 27 (2003) 237-251.

[20] R. Oberlin, B. Street and R.S. Strichartz, Sampling on the Sierpinski gasket. Experiment. Math. 12 (2003) 403-418. | Zbl

[21] J. Rauch, Partial differential equations. Graduate Texts in Mathematics 128, Springer-Verlag, New York (1991). | MR | Zbl

[22] C. Sabot, Spectral properties of self-similar lattices and iteration of rational maps. Mém. Soc. Math. Fr. (N.S.) 92 (2003) vi+104. | Numdam | MR | Zbl

[23] C. Sabot, Electrical networks, symplectic reductions, and application to the renormalization map of self-similar lattices, in Fractal geometry and applications: a jubilee of Benoît Mandelbrot. Part 1, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 72 (2004) 155-205. | Zbl

[24] B. Sapoval and T. Gobron, Vibration of strongly irregular fractal resonators. Phys. Rev. E 47 (1993).

[25] B. Sapoval, T. Gobron and A. Margolina, Vibration of fractal drums. Phys. Rev. Lett. 67 (1991).