The Spectrum of the Damped Wave Operator for a Bounded Domain in<i>R</i><sup>2</sup>

Asch, Mark; Lebeau, Gilles

doi:10.1080/10586458.2003.10504494

Cited by 35 publications

(54 citation statements)

References 8 publications

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“…In the situation where b is a characteristic function of a strip of the torus (hence discontinuous), we provide several numerical simulations (in the spirit of [4]) showing that the decay rate should be of type , proved in the first part of the paper in a very general setting.…”

Section: Vi-4mentioning

confidence: 99%

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Some decay properties for the damped wave equation on the torus

Anantharaman¹,

Léautaud²

2013

Journées Équations Aux Dérivées Partielles

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Quelques propriétés de décroissance pour l'équation des ondes amorties sur le tore RésuméCet article est la version courte d'un travail en cours [1], et a fait l'objet d'un exposé du second auteur au cours des Journées "Équations aux Dérivées Partielles" (Biarritz, 2012).On s'intéresse aux taux de décroissance de l'énergie pour l'équation des ondes amorties dans des situations où le coefficient d'amortissement b ne satisfait pas la condition de contrôle géométrique. On donne tout d'abord un lien avec la contrôlabilité de l'équation de Schrödinger associée. On montre que l'observabilité du groupe de Schrödinger implique la décroissance à taux 1/ √ t du semigroupe des ondes amorties (taux meilleur que le taux logarithmique a priori fourni par le théorème de Lebeau).Dans un second temps, on se focalise sur le tore 2-D. Toujours en supposant que le contrôle géométrique n'est pas réalisé, on montre que le semigroupe décroît au mieux à taux 1/t. Réciproquement, pour des coefficients d'amortissements b réguliers, on prouve la décroissance à taux 1/t 1−ε , pour tout ε > 0.Dans le cas où le le coefficient d'amortissement est la fonction caractéristique d'une bande (donc discontinu), on effectue des simulations numériques qui semblent exhiber un taux de décroissance strictement pire que 1/t.En particulier, notre étude tend à montrer que le taux de décroissance dépend fortement du taux d'annulation de b. VI-1 AbstractThis article is a proceedings version of the ongoing work [1], and has been the object of a talk of the second author during the Journées "Équations aux Dérivées Partielles" (Biarritz, 2012).We address the decay rates of the energy of the damped wave equation when the damping coefficient b does not satisfy the Geometric Control Condition (GCC). First, we give a link with the controllability of the associated Schrödinger equation. We prove that the observability of the Schrödinger group implies that the semigroup associated to the damped wave equation decays at rate 1/ √ t (which is a stronger rate than the general logarithmic one predicted by the Lebeau Theorem).Second, we focus on the 2-dimensional torus. We prove that the best decay one can expect is 1/t, as soon as the damping region does not satisfy GCC. Conversely, for smooth damping coefficients b, we show that the semigroup decays at rate 1/t 1−ε , for all ε > 0.In the case where the damping coefficient is a characteristic function of a strip (hence discontinuous), we give numerical evidence of decay rates strictly worse than 1/t. In particular, our study tends to prove that the decay rate highly depends on the way b vanishes.

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Section: Vi-4mentioning

confidence: 99%

“…The particular shape of the spectrum (see [4] and Figures 4.1 and 4.4), separated in several branches, is very helpful to obtain precise estimates.…”

Section: Vi-8mentioning

confidence: 99%

Some decay properties for the damped wave equation on the torus

Anantharaman¹,

Léautaud²

2013

Journées Équations Aux Dérivées Partielles

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“…(21) which differs from the most classical finite-difference scheme by the addition of viscous terms. We shall see below that this numerical approximation scheme converges in the energy space to the continuous damped wave equation (1).…”

Section: Finite Difference Schemementioning

confidence: 99%

“…The objects corresponding to this example are summarized on (3,1), (3,5), (7,3), (9, 0), (9, 1), (9, 2), (9, 4), (9, 5)}, (3,5), (4,5), (7,3), (8,3), (9,3), (1,6), (2,6), (5,6), (6,6), (7,6), (8,6)}.…”

Section: Remark 12mentioning

confidence: 99%

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Uniform stabilization of a viscous numerical approximation for a locally damped wave equation

Münch

Pazoto

2007

ESAIM: COCV

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Abstract. This work is devoted to the analysis of a viscous finite-difference space semi-discretization of a locally damped wave equation in a regular 2-D domain. The damping term is supported in a suitable subset of the domain, so that the energy of solutions of the damped continuous wave equation decays exponentially to zero as time goes to infinity. Using discrete multiplier techniques, we prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size) exponential decay of the energy for the solutions of the numerical scheme. The numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept, which guarantees that the low frequencies are damped correctly. Numerical experiments are presented and confirm these theoretical results. These results extend those by Tcheugoué-Tébou and Zuazua [Numer. Math. 95, 563-598 (2003)] where the 1-D case was addressed as well the square domain in 2-D. The methods and results in this paper extend to smooth domains in any space dimension.Mathematics Subject Classification. 65M06.

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