This paper is devoted to some elliptic boundary value problems in a self-similar ramified domain of 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.
Keywords: domains with fractal boundaries, Helmholtz equation, Neumann boundary conditions, transparent boundary conditions
@article{M2AN_2006__40_4_623_0, author = {Achdou, Yves and Sabot, Christophe and Tchou, Nicoletta}, title = {Diffusion and propagation problems in some ramified domains with a fractal boundary}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {623--652}, publisher = {EDP-Sciences}, volume = {40}, number = {4}, year = {2006}, doi = {10.1051/m2an:2006027}, mrnumber = {2274772}, zbl = {1112.65115}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an:2006027/} }
TY - JOUR AU - Achdou, Yves AU - Sabot, Christophe AU - Tchou, Nicoletta TI - Diffusion and propagation problems in some ramified domains with a fractal boundary JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2006 SP - 623 EP - 652 VL - 40 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2006027/ DO - 10.1051/m2an:2006027 LA - en ID - M2AN_2006__40_4_623_0 ER -
%0 Journal Article %A Achdou, Yves %A Sabot, Christophe %A Tchou, Nicoletta %T Diffusion and propagation problems in some ramified domains with a fractal boundary %J ESAIM: Modélisation mathématique et analyse numérique %D 2006 %P 623-652 %V 40 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an:2006027/ %R 10.1051/m2an:2006027 %G en %F M2AN_2006__40_4_623_0
Achdou, Yves; Sabot, Christophe; Tchou, Nicoletta. Diffusion and propagation problems in some ramified domains with a fractal boundary. ESAIM: Modélisation mathématique et analyse numérique, Volume 40 (2006) no. 4, pp. 623-652. doi : 10.1051/m2an:2006027. http://archive.numdam.org/articles/10.1051/m2an:2006027/
[1] A multiscale numerical method for Poisson problems in some ramified domains with a fractal boundary. SIAM Multiscale Model. Simul. (2006) (accepted for publication). | MR
, and ,[2] Transparent boundary conditions for Helmholtz equation in some ramified domains with a fractal boundary. J. Comput. Phys. (2006) (in press). | MR | Zbl
, and ,[3] Sobolev spaces. Academic Press, New York-London (1975). Pure Appl. Math. 65. | MR | Zbl
,[4] Analyse fonctionnelle. Collection Mathématiques Appliquées pour la Maîtrise. Théorie et applications. Masson, Paris, 1983. | MR | Zbl
,[5] Physique du transport diffusif de l'oxygène dans le poumon humain. Ph.D. thesis, École Polytechnique (2003).
,[6] The finite element method on the Sierpinski gasket. Constr. Approx. 17 (2001) 561-588. | Zbl
, and ,[7] Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics 24, Pitman (Advanced Publishing Program), Boston, MA (1985). | MR | Zbl
,[8] Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981) 713-747. | Zbl
,[9] Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147 (1981) 71-88. | Zbl
,[10] Function spaces on subsets of . Math. Rep. 2 (1984) xiv+221. | MR | Zbl
and ,[11] Exact nonreflecting boundary conditions. J. Comput. Phys. 82 (1989) 172-192. | Zbl
and ,[12] A transmission problem with a fractal interface. Z. Anal. Anwendungen 21 (2002) 113-133. | Zbl
,[13] Second order transmission problems across a fractal surface. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 27 (2003) 191-213.
,[14] The fractal geometry of nature. Freeman and Co (1982). | MR | Zbl
,[15] Interplay between flow distribution and geometry in an airway tree. Phys. Rev. Lett. 90 (2003).
, , and ,[16] The optimal bronchial tree is dangerous. Nature 427 (2004) 633-636.
, , and ,[17] Sobolev spaces. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin (1985). Translated from the Russian by T.O. Shaposhnikova. | Zbl
,[18] Energy functionals on certain fractal structures. J. Convex Anal. 9 (2002) 581-600. | Zbl
,[19] Variational problems with fractal layers. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 27 (2003) 237-251.
and ,[20] Sampling on the Sierpinski gasket. Experiment. Math. 12 (2003) 403-418. | Zbl
, and ,[21] Partial differential equations. Graduate Texts in Mathematics 128, Springer-Verlag, New York (1991). | MR | Zbl
,[22] 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] 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] Vibration of strongly irregular fractal resonators. Phys. Rev. E 47 (1993).
and ,[25] Vibration of fractal drums. Phys. Rev. Lett. 67 (1991).
, and ,Cited by Sources: