# 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

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topInfante, 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 -

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