Boundary observability for the space semi-discretizations of the 1 – d wave equation

Juan Antonio Infante; Enrique Zuazua

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 33, Issue: 2, page 407-438
  • ISSN: 0764-583X

Abstract

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We consider space semi-discretizations of the 1-d wave equation in a bounded interval with homogeneous Dirichlet boundary conditions. We analyze the problem of boundary observability, i.e., the problem of whether the total energy of solutions can be estimated uniformly in terms of the energy concentrated on the boundary as the net-spacing h → 0. We prove that, due to the spurious modes that the numerical scheme introduces at high frequencies, there is no such a uniform bound. We prove however a uniform bound in a subspace of solutions generated by the low frequencies of the discrete system. When h → 0 this finite-dimensional spaces increase and eventually cover the whole space. We thus recover the well-known observability property of the continuous system as the limit of discrete observability estimates as the mesh size tends to zero. We consider both finite-difference and finite-element semi-discretizations.

How to cite

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Infante, Juan Antonio, and Zuazua, Enrique. "Boundary observability for the space semi-discretizations of the 1 – d wave equation." ESAIM: Mathematical Modelling and Numerical Analysis 33.2 (2010): 407-438. <http://eudml.org/doc/197593>.

@article{Infante2010,
abstract = { We consider space semi-discretizations of the 1-d wave equation in a bounded interval with homogeneous Dirichlet boundary conditions. We analyze the problem of boundary observability, i.e., the problem of whether the total energy of solutions can be estimated uniformly in terms of the energy concentrated on the boundary as the net-spacing h → 0. We prove that, due to the spurious modes that the numerical scheme introduces at high frequencies, there is no such a uniform bound. We prove however a uniform bound in a subspace of solutions generated by the low frequencies of the discrete system. When h → 0 this finite-dimensional spaces increase and eventually cover the whole space. We thus recover the well-known observability property of the continuous system as the limit of discrete observability estimates as the mesh size tends to zero. We consider both finite-difference and finite-element semi-discretizations. },
author = {Infante, Juan Antonio, Zuazua, Enrique},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Wave equation; semi-discretization; observability.; wave equation; finite difference; finite element; boundary observability},
language = {eng},
month = {3},
number = {2},
pages = {407-438},
publisher = {EDP Sciences},
title = {Boundary observability for the space semi-discretizations of the 1 – d wave equation},
url = {http://eudml.org/doc/197593},
volume = {33},
year = {2010},
}

TY - JOUR
AU - Infante, Juan Antonio
AU - Zuazua, Enrique
TI - Boundary observability for the space semi-discretizations of the 1 – d wave equation
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 33
IS - 2
SP - 407
EP - 438
AB - We consider space semi-discretizations of the 1-d wave equation in a bounded interval with homogeneous Dirichlet boundary conditions. We analyze the problem of boundary observability, i.e., the problem of whether the total energy of solutions can be estimated uniformly in terms of the energy concentrated on the boundary as the net-spacing h → 0. We prove that, due to the spurious modes that the numerical scheme introduces at high frequencies, there is no such a uniform bound. We prove however a uniform bound in a subspace of solutions generated by the low frequencies of the discrete system. When h → 0 this finite-dimensional spaces increase and eventually cover the whole space. We thus recover the well-known observability property of the continuous system as the limit of discrete observability estimates as the mesh size tends to zero. We consider both finite-difference and finite-element semi-discretizations.
LA - eng
KW - Wave equation; semi-discretization; observability.; wave equation; finite difference; finite element; boundary observability
UR - http://eudml.org/doc/197593
ER -

Citations in EuDML Documents

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  1. Karim Ramdani, Takéo Takahashi, Marius Tucsnak, Uniformly exponentially stable approximations for a class of second order evolution equations
  2. Liliana León, Enrique Zuazua, Boundary controllability of the finite-difference space semi-discretizations of the beam equation
  3. Michel Mehrenberger, Paola Loreti, An Ingham type proof for a two-grid observability theorem
  4. Paola Loreti, Michel Mehrenberger, An Ingham type proof for a two-grid observability theorem
  5. Liliana León, Enrique Zuazua, Boundary controllability of the finite-difference space semi-discretizations of the beam equation
  6. Arnaud Münch, A uniformly controllable and implicit scheme for the 1-D wave equation
  7. Arnaud Münch, A uniformly controllable and implicit scheme for the 1-D wave equation
  8. Mark Asch, Marion Darbas, Jean-Baptiste Duval, Numerical solution of an inverse initial boundary value problem for the wave equation in the presence of conductivity imperfections of small volume
  9. Romain Joly, Perturbation de la dynamique des équations des ondes amorties
  10. Arnaud Münch, Ademir Fernando Pazoto, Uniform stabilization of a viscous numerical approximation for a locally damped wave equation

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