Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square

Castro Barbero, Carlos Manuel and Micu, Sorín and Münch, Arnaud (2008). Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square. "Ima Journal of Numerical Analysis", v. 28 (n. 1); pp. 186-214. ISSN 0272-4979. https://doi.org/10.1093/imanum/drm012.

Description

Title: Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square
Author/s:
  • Castro Barbero, Carlos Manuel
  • Micu, Sorín
  • Münch, Arnaud
Item Type: Article
Título de Revista/Publicación: Ima Journal of Numerical Analysis
Date: January 2008
ISSN: 0272-4979
Volume: 28
Subjects:
Freetext Keywords: exact controllability; observability; numerical approximation of controls; wave equation
Faculty: E.T.S.I. Caminos, Canales y Puertos (UPM)
Department: Matemática e Informática Aplicadas a la Ingeniería Civil [hasta 2014]
Creative Commons Licenses: Recognition - No derivative works - Non commercial

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Abstract

This paper studies the numerical approximation of the boundary control for the wave equation in a square domain. It is known that the discrete and semi-discrete models ob- tained by discretizing the wave equation with the usual ¯nite di®erence or ¯nite element methods do not provide convergent sequences of approximations to the boundary control of the continuous wave equation, as the mesh size goes to zero (see [7, 15]). Here we introduce and analyze a new semi-discrete model based on the space discretization of the wave equa- tion using a mixed ¯nite element method with two di®erent basis functions for the position and velocity. The main theoretical result is a uniform observability inequality which allows us to construct a sequence of approximations converging to the minimal L2¡norm control of the continuous wave equation. We also introduce a fully-discrete system, obtained from our semi-discrete scheme, for which we conjecture that it provides a convergent sequence of discrete approximations as both h and ¢t, the time discretization parameter, go to zero. We illustrate this fact with several numerical experiments.

More information

Item ID: 2922
DC Identifier: http://oa.upm.es/2922/
OAI Identifier: oai:oa.upm.es:2922
DOI: 10.1093/imanum/drm012
Official URL: http://imajna.oxfordjournals.org/content/vol28/issue1/index.dtl
Deposited by: Memoria Investigacion
Deposited on: 14 May 2010 11:45
Last Modified: 20 Apr 2016 12:32
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