Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method

Castro, Carlos; Micu, Sorin

doi:10.1007/s00211-005-0651-0

Cited by 89 publications

(99 citation statements)

References 16 publications

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“…This part is inspired in [5,6] where similar results have been derived for uniform meshes. We consider a mesh S n as in (1.6) and derive an approximation scheme for (3.1) from a mixed finite element method.…”

Section: The Semi-discrete Settingmentioning

confidence: 93%

“…The proof of Lemma 3.1 is the same as in [5]. For completeness, we will give a sketch of the proof hereafter.…”

Section: Lemma 31 For Any Integer N the Functional J N Is Strictlymentioning

confidence: 99%

“…On S n , the mixed finite element approximation scheme for system (1.1) reads as (see [7,15] or [5]):…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, as in the continuous setting, Theorem 1.2 has several applications to controllability and stabilization properties for the space semi-discrete 1d wave equations (1.7). In Section 3, similarly as in [5], using precisely the same duality as in the continuous case, we present an application to the boundary null controllability of the space semi-discrete approximation scheme of the 1d wave equation. Later, in Section 4, following [1], we study the decay properties of the energy for semi-discrete approximation schemes of 1d damped wave equations.…”

mentioning

confidence: 99%

“…Let us also mention the results [10,[26][27][28] that amounts to adding an extra term in (1.12) which is non-negligible only for the high frequencies. A last possible cure was proposed in [1,15] and later analyzed in [5]: a 1d semi-discrete scheme derived from a mixed finite element method was proposed, which has the property that the group velocity of the waves is bounded from below. Also note that an extension of [5] to the 2d case in the square was proposed in [6].…”

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confidence: 99%

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Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes

Ervedoza

2008

ESAIM: COCV

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Abstract. The goal of this article is to analyze the observability properties for a space semi-discrete approximation scheme derived from a mixed finite element method of the 1d wave equation on nonuniform meshes. More precisely, we prove that observability properties hold uniformly with respect to the mesh-size under some assumptions, which, roughly, measures the lack of uniformity of the meshes, thus extending the work [Castro and Micu, Numer. Math. 102 (2006) 413-462] to nonuniform meshes. Our results are based on a precise description of the spectrum of the discrete approximation schemes on nonuniform meshes, and the use of Ingham's inequality. We also mention applications to the boundary null controllability of the 1d wave equation, and to stabilization properties for the 1d wave equation. We finally present some applications for the corresponding fully discrete schemes, based on recent articles by the author.Mathematics Subject Classification. 35L05, 35P20, 47A75, 93B05, 93B07, 93B60, 93D15.

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Section: The Semi-discrete Settingmentioning

confidence: 93%

“…The proof of Lemma 3.1 is the same as in [5]. For completeness, we will give a sketch of the proof hereafter.…”

Section: Lemma 31 For Any Integer N the Functional J N Is Strictlymentioning

confidence: 99%

“…On S n , the mixed finite element approximation scheme for system (1.1) reads as (see [7,15] or [5]):…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes

Ervedoza

2008

ESAIM: COCV

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Uniform exponential decay of the energy for a fully discrete wave equation with point‐wise dissipation

Amara

2023

Math Methods in App Sciences

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In this paper, the uniform exponential decay of the energy for a fully discretization of the 1-D wave equation has been studied with a punctual interior damping at 𝜉 (𝜉 ∈ (0, 1)). It is known that the usual scheme obtained with a space discretization is not uniformly exponential decaying. In this direction, we have introduced an implicit finite difference scheme that differs from the usual centered one by additional terms of order h 2 and 𝛿t 2 (where h and 𝛿t represent the discretization parameters). In this approach, the fully discrete system was decomposed into two subsystems, namely conservative and nonconservative.According to the literature, it has been noted that under a numerical hypothesis on 𝜉, a uniform observability inequality holds for a conservative system. The uniform exponential decay of the energy for the damped system was proved via the application of the observability inequality.

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On the observability inequalities of time discrete 2‐D integro‐differential systems in square domains

2020

Numerical Methods Partial

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In this article, we consider an observability inequality of time discrete approximation schemes for a class of integro‐differential equations in square domains. The equation is discretized in time by the back‐ward Euler method in combination with Lubich's convolution quadrature. We derive uniform observability inequality for time discretization schemes in which the high frequency components have been filtered. In this way, the well‐known exact observability inequality of the integro‐differential systems (see Loreti and Sforza, Evolution Equations and Control Theory, 2018, Vol. 7, No. 1, pp. 61–77, Thm. 1.2) can be reproduced as the limit, as the time step Δt → 0. The time discrete observability inequality is established by means of a time‐discrete version of the classical harmonic analysis approach. Numerical experiments are presented to show the theoretical analysis.

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Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method

Cited by 89 publications

References 16 publications

Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes

Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes

Uniform exponential decay of the energy for a fully discrete wave equation with point‐wise dissipation

On the observability inequalities of time discrete 2‐D integro‐differential systems in square domains

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