# American Institute of Mathematical Sciences

June  2019, 14(2): 205-264. doi: 10.3934/nhm.2019010

## Wave propagation in fractal trees. Mathematical and numerical issues

 1 POEMS (UMR 7231 CNRS-INRIA-ENSTA), ENSTA ParisTech, 828 Boulevard des Maréchaux, Palaiseau, F-91120, France 2 Technische Universität Darmstadt, Fachgebiet Mathematik, AG Numerik und Wissenschaftliches Rechnen, Dolivostraße 15, Darmstadt, D-64293, Germany

Received  December 2016 Revised  October 2018 Published  April 2019

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.

Citation: Patrick Joly, Maryna Kachanovska, Adrien Semin. Wave propagation in fractal trees. Mathematical and numerical issues. Networks & Heterogeneous Media, 2019, 14 (2) : 205-264. doi: 10.3934/nhm.2019010
##### References:

show all references

##### References:
Left: the limit tree $\mathbb{G}$. Right: the thick tree ${{\mathbb{G}}^{\delta }}$
General tree. We numbered here the edges. We plotted in red the subtree $\mathcal{T}_{2, 4}$ and in blue the truncated tree $\mathcal{T}^1$
"1D tree" corresponding to the case $\alpha$ = 0.5
Example of p-adic tree for p = 2. Left: iterative construction. Right: weight repartition
Inductive construction of the mesh $\Gamma_n$
A summary of the results of sections 3.1-3.3
Polar mesh of the quarter plane
Plots of $|\mathbf{\Lambda}_\mathfrak{d}(\omega)|$ (left) and $|\mathbf{\Lambda}_\mathfrak{n}(\omega)|$ (right), for $|\omega| < 2\pi$, $\alpha = \mu = 0.6$
Plots of $\Im\left(\omega^{-1}\mathbf{\Lambda}_\mathfrak{d}(\omega)\right)$ (left) and $\Im\left(\omega^{-1}\mathbf{\Lambda}_\mathfrak{n}(\omega)\right)$ (right), for $|\omega| < 2\pi$, $\alpha = \mu = 0.6$. Remark that $\omega^{-1}\mathbf{\Lambda}_\mathfrak{d}(\omega)$ has a pole in $\omega = 0$, unlike $\omega^{-1}\mathbf{\Lambda}_\mathfrak{n}(\omega)$
Plots of $\left|\boldsymbol{\Lambda}_D(\omega)\right|$ (left) for $\alpha = 0.6$, $\mu = 0.2$ and of $\left|\boldsymbol{\Lambda}_N(\omega)\right|$ (right) for $\alpha = 0.6$, $\mu = 2$
Left row: the dependence of $u(M, t)$ on time for the exact (red solid line) and the truncated tree on 7 generations (blue dashed line). Top: Dirichlet condition. Middle: the first order DtN condition. Bottom: the second order DtN condition.
Right row: the dependence of $u(M, t)$ on time for the exact (red solid line) and the truncated tree on 9 generations (blue dashed line). Top: Dirichlet condition. Middle: the first order DtN condition. Bottom: the second order DtN condition
${{\text{L}}^{2}}$-error between exact and approximate solutions, with respect to the number of generations and the order of the approximate boundary condition
L2-error between the exact and approximate solutions, with respect to the number of generations and the order of the approximate boundary condition
 Number of generations $n+1$ Dirichlet condition First order condition Second order condition Gain with first order Gain with second order $5$ $0.429$ $0.320$ $1.23\times10^{-1}$ 1.34 3.05 $6$ $0.370$ $0.205$ $5.01\times10^{-2}$ 1.80 7.35 $7$ $0.217$ $0.075$ $1.37\times10^{-2}$ 2.89 15.83 $8$ $0.083$ $0.018$ $2.72\times10^{-3}$ 4.53 30.5 $9$ $0.023$ $0.0031$ $3.84\times10^{-4}$ 7.47 59.9
 Number of generations $n+1$ Dirichlet condition First order condition Second order condition Gain with first order Gain with second order $5$ $0.429$ $0.320$ $1.23\times10^{-1}$ 1.34 3.05 $6$ $0.370$ $0.205$ $5.01\times10^{-2}$ 1.80 7.35 $7$ $0.217$ $0.075$ $1.37\times10^{-2}$ 2.89 15.83 $8$ $0.083$ $0.018$ $2.72\times10^{-3}$ 4.53 30.5 $9$ $0.023$ $0.0031$ $3.84\times10^{-4}$ 7.47 59.9
 [1] S. L. Ma'u, P. Ramankutty. An averaging method for the Helmholtz equation. Conference Publications, 2003, 2003 (Special) : 604-609. doi: 10.3934/proc.2003.2003.604 [2] Kim Dang Phung. Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1057-1093. doi: 10.3934/dcds.2008.20.1057 [3] Simone Creo, Maria Rosaria Lancia, Alejandro Vélez-Santiago, Paola Vernole. Approximation of a nonlinear fractal energy functional on varying Hilbert spaces. Communications on Pure & Applied Analysis, 2018, 17 (2) : 647-669. doi: 10.3934/cpaa.2018035 [4] John Sylvester. An estimate for the free Helmholtz equation that scales. Inverse Problems & Imaging, 2009, 3 (2) : 333-351. doi: 10.3934/ipi.2009.3.333 [5] Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285 [6] Harbir Antil, Mahamadi Warma. Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence. Mathematical Control & Related Fields, 2019, 9 (1) : 1-38. doi: 10.3934/mcrf.2019001 [7] Sang-Yeun Shim, Marcos Capistran, Yu Chen. Rapid perturbational calculations for the Helmholtz equation in two dimensions. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 627-636. doi: 10.3934/dcds.2007.18.627 [8] Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343 [9] Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1273-1295. doi: 10.3934/dcdsb.2019016 [10] Roman Chapko, B. Tomas Johansson. An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions. Inverse Problems & Imaging, 2008, 2 (3) : 317-333. doi: 10.3934/ipi.2008.2.317 [11] Yoshitsugu Kabeya. Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3529-3559. doi: 10.3934/dcds.2020040 [12] Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105 [13] Gleb G. Doronin, Nikolai A. Larkin. Kawahara equation in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 783-799. doi: 10.3934/dcdsb.2008.10.783 [14] Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations & Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325 [15] Chao Zhang, Xia Zhang, Shulin Zhou. Gradient estimates for the strong $p(x)$-Laplace equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4109-4129. doi: 10.3934/dcds.2017175 [16] Yutian Lei, Congming Li, Chao Ma. Decay estimation for positive solutions of a $\gamma$-Laplace equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 547-558. doi: 10.3934/dcds.2011.30.547 [17] Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013 [18] Andrei Fursikov, Lyubov Shatina. Nonlocal stabilization by starting control of the normal equation generated by Helmholtz system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1187-1242. doi: 10.3934/dcds.2018050 [19] Wenjia Jing, Olivier Pinaud. A backscattering model based on corrector theory of homogenization for the random Helmholtz equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5377-5407. doi: 10.3934/dcdsb.2019063 [20] Roberto A. Capistrano-Filho, Shuming Sun, Bing-Yu Zhang. General boundary value problems of the Korteweg-de Vries equation on a bounded domain. Mathematical Control & Related Fields, 2018, 8 (3&4) : 583-605. doi: 10.3934/mcrf.2018024

2018 Impact Factor: 0.871

## Metrics

• HTML views (327)
• Cited by (0)

• on AIMS