Wave propagation in fractal trees. Mathematical and numerical issues. (English) Zbl 1427.35018
Summary: We propose and analyze a mathematical model for wave propagation in infinite trees with self-similar structure at infinity. This emphasis is put on the construction and approximation of transparent boundary conditions. The performance of the constructed boundary conditions is then illustrated by numerical experiments.
MSC:
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35L05 | Wave equation |
| 35R02 | PDEs on graphs and networks (ramified or polygonal spaces) |
References:
| [1] | Y. Achdou; F. Camilli; A. Cutrì; N. Tchou, Hamilton-Jacobi equations constrained on networks, Nonlinear Differential Equations and Applications NoDEA, 20, 413-445 (2013) · Zbl 1268.35120 · doi:10.1007/s00030-012-0158-1 |
| [2] | Y. Achdou; C. Sabot; N. Tchou, Diffusion and propagation problems in some ramified domains with a fractal boundary, ESAIM: Mathematical Modelling and Numerical Analysis, 40, 623-652 (2006) · Zbl 1112.65115 · doi:10.1051/m2an:2006027 |
| [3] | Y. Achdou; C. Sabot; N. Tchou, Transparent boundary conditions for a class of boundary value problems in some ramified domains with a fractal boundary, C. R. Math. Acad. Sci. Paris, 342, 605-610 (2006) · Zbl 1096.65113 · doi:10.1016/j.crma.2006.02.024 |
| [4] | Y. Achdou; C. Sabot; N. Tchou, Transparent boundary conditions for the Helmholtz equation in some ramified domains with a fractal boundary, J. Comput. Phys., 220, 712-739 (2007) · Zbl 1109.65096 · doi:10.1016/j.jcp.2006.05.033 |
| [5] | Y. Achdou; N. Tchou, Boundary value problems with nonhomogeneous Neumann conditions on a fractal boundary, C. R. Math. Acad. Sci. Paris, 342, 611-616 (2006) · Zbl 1096.65114 · doi:10.1016/j.crma.2006.02.025 |
| [6] | Y. Achdou; N. Tchou, Neumann conditions on fractal boundaries, Asymptot. Anal., 53, 61-82 (2007) · Zbl 1148.35017 |
| [7] | Y. Achdou and N. Tchou, Boundary value problems in ramified domains with fractal boundaries, In Domain decomposition methods in science and engineering XVII, volume 60 of Lect. Notes Comput. Sci. Eng., pages 419-426. Springer, Berlin, 2008. · Zbl 1139.65320 |
| [8] | F. Ali Mehmeti; S. Nicaise, Nonlinear interaction problems, Nonlinear Anal., 20, 27-61 (1993) · Zbl 0817.35035 · doi:10.1016/0362-546X(93)90183-S |
| [9] | W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, volume 96 of Monographs in Mathematics, Birkhäuser/Springer Basel AG, Basel, second edition, 2011. · Zbl 1226.34002 |
| [10] | L. Banjai; C. Lubich; F.-J. Sayas, Stable numerical coupling of exterior and interior problems for the wave equation, Numer. Math., 129, 611-646 (2015) · Zbl 1314.65122 · doi:10.1007/s00211-014-0650-0 |
| [11] | G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, volume 186 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013. · Zbl 1318.81005 |
| [12] | J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., Inc., New York, 1976. · Zbl 1226.05083 |
| [13] | P. Cazeaux; C. Grandmont; Y. Maday, Homogenization of a model for the propagation of sound in the lungs, Multiscale Model. Simul., 13, 43-71 (2015) · doi:10.1137/130916576 |
| [14] | S. M. Cioabă and M. Ram Murty, A First Course in Graph Theory and Combinatorics, volume 55 of Texts and Readings in Mathematics, Hindustan Book Agency, New Delhi, 2009. · Zbl 1216.05001 |
| [15] | B. Engquist; A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31, 629-651 (1977) · Zbl 0367.65051 · doi:10.1090/S0025-5718-1977-0436612-4 |
| [16] | A. Georgakopoulos, S. Haeseler, M. Keller, D. Lenz and R. K. Wojciechowski, Graphs of finite measure, J. Math. Pures Appl. (9), 103 (2015), 1093-1131. · Zbl 1314.31017 |
| [17] | F. Gesztesy; E. Tsekanovskii, On matrix-valued Herglotz functions, Math. Nachr., 218, 61-138 (2000) · Zbl 0961.30027 · doi:10.1002/1522-2616(200010)218:13.0.CO;2-D |
| [18] | P. Joly and A. Semin, Construction and analysis of improved Kirchoff conditions for acoustic wave propagation in a junction of thin slots, In ESAIM: Proceedings, volume 25, pages 44-67. EDP Sciences, 2008. · Zbl 1156.76422 |
| [19] | T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York, 1966. · Zbl 0148.12601 |
| [20] | J. L. Kelley, General Topology, Springer-Verlag, New York-Berlin, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathematics, No. 27. · Zbl 0306.54002 |
| [21] | P. Kuchment, Graph models for waves in thin structures, Waves in Random Media, 12 (2002), R1-R24. · Zbl 1063.35525 |
| [22] | B. B. Mandelbrot, The Fractal Geometry of Nature, San Francisco, Calif., 1982. · Zbl 0504.28001 |
| [23] | B. Maury; D. Salort; C. Vannier, Trace theorems for trees, application to the human lungs, Network and Heteregeneous Media, 4, 469-500 (2009) · Zbl 1221.05054 · doi:10.3934/nhm.2009.4.469 |
| [24] | K. Naimark and M. Solomyak, Eigenvalue estimates for the weighted Laplacian on metric trees, Proc. London Math. Soc. (3), 80 (2000), 690-724. · Zbl 1046.34092 |
| [25] | K. Naimark; M. Solomyak, Geometry of Sobolev spaces on regular trees and the Hardy inequalities, Russ. J. Math. Phys., 8, 322-335 (2001) · Zbl 1187.46028 |
| [26] | S. Nicaise, Elliptic operators on elementary ramified spaces, Integral Equations Operator Theory, 11, 230-257 (1988) · Zbl 0649.35027 · doi:10.1007/BF01272120 |
| [27] | H. Pasterkamp; S. S. Kraman; G. R. Wodicka, Respiratory sounds: advances beyond the stethoscope, American journal of respiratory and critical care medicine, 156, 974-987 (1997) · doi:10.1164/ajrccm.156.3.9701115 |
| [28] | N. Pozin, S. Montesantos, I. Katz, M. Pichelin, I. Vignon-Clementel and C. Grandmont, A tree-parenchyma coupled model for lung ventilation simulation, Int. J. Numer. Methods Biomed. Eng., 33 (2017), e2873, 30pp. |
| [29] | J. Rubinstein; M. Schatzman, Variational problems on multiply connected thin strips. I. Basic estimates and convergence of the Laplacian spectrum, Arch. Ration. Mech. Anal., 160, 271-308 (2001) · Zbl 0997.49003 · doi:10.1007/s002050100164 |
| [30] | J. Rubinstein; M. Schatzman, Variational problems on multiply connected thin strips. â…¡. Convergence of the Ginzburg-Landau functional, Arch. Ration. Mech. Anal., 160, 309-324 (2001) · Zbl 0997.49004 · doi:10.1007/s002050100165 |
| [31] | D. Rueter; H.-P. Hauber; D. Droeman; P. Zabel; S. Uhlig, Low-frequency ultrasound permeates the human thorax and lung: A novel approach to non-invasive monitoring, Ultraschall in der Medizin-European Journal of Ultrasound, 31, 53-62 (2010) |
| [32] | A. Semin, Propagation d’ondes dans des jonctions de fentes minces, PhD thesis, Université de Paris-Sud 11, 2010. |
| [33] | M. Solomyak, Laplace and Schrödinger operators on regular metric trees: The discrete spectrum case, In Function Spaces, Differential Operators and Nonlinear Analysis (Teistungen, 2001), pages 161-181. Birkhäuser, Basel, 2003. · Zbl 1079.34065 |
| [34] | M. Solomyak, On approximation of functions from Sobolev spaces on metric graphs, J. Approx. Theory, 121, 199-219 (2003) · Zbl 1027.46037 · doi:10.1016/S0021-9045(03)00033-9 |
| [35] | M. Solomyak, On the spectrum of the laplacian on regular metric trees, Waves in Random Media, 14 (2004), S155-S171. · Zbl 1077.47513 |
| [36] | O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems, Springer, New York, 2008. Finite and boundary elements, Translated from the 2003 German original. · Zbl 1130.65001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
