# 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
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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
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