[Solutions localisées et mécanismes de filtrage pour les approximations de Galerkin discontinues de l'équation des ondes]
On développe une analyse de Fourier complète de l'équation des ondes unidimensionnelle semi-discrétisée en espace obtenue dans l'approximation numérique de l'équation des ondes par une méthode de Galerkin discontinue (GD) dans un maillage uniforme. On met en évidence la coexistence de deux composantes dans le système numérique : une physique, et une parasite liée aux discontinuités que la solution numérique permet. Chaque relation de dispersion contient des points critiques où la vitesse de groupe correspondante s'annule. En suivant les constructions faites antérieurement pour le schéma en différences finies, on construit d'une manière rigoureuse des paquets d'ondes qui se propagent à une vitesse arbitrairement petite, concentrés soit sur l'une, soit sur l'autre de ces deux composantes. On développe aussi des mécanismes de filtrage permettant de récupérer les propriétés de propagation des solutions de l'équation continue. Enfin, on présente une application à l'approximation numérique des problèmes de contrôle.
We perform a complete Fourier analysis of the semi-discrete wave equation obtained through a discontinuous Galerkin (DG) approximation of the continuous wave equation on an uniform grid. The resulting system exhibits the interaction of two types of components: a physical one and a spurious one, related to the possible discontinuities that the numerical solution allows. Each dispersion relation contains critical points where the corresponding group velocity vanishes. Following previous constructions, we rigorously build wave packets with arbitrarily small velocity of propagation concentrated either on the physical or on the spurious component. We also develop filtering mechanisms aimed at recovering the uniform velocity of propagation of the continuous solutions. Finally, some applications to numerical approximation issues of control problems are also presented.
Accepté le :
Publié le :
@article{CRMATH_2010__348_19-20_1087_0,
author = {Marica, Aurora and Zuazua, Enrique},
title = {Localized solutions and filtering mechanisms for the discontinuous {Galerkin} semi-discretizations of the $ 1-d$ wave equation},
journal = {Comptes Rendus. Math\'ematique},
pages = {1087--1092},
publisher = {Elsevier},
volume = {348},
number = {19-20},
year = {2010},
doi = {10.1016/j.crma.2010.09.012},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2010.09.012/}
}
TY - JOUR AU - Marica, Aurora AU - Zuazua, Enrique TI - Localized solutions and filtering mechanisms for the discontinuous Galerkin semi-discretizations of the $ 1-d$ wave equation JO - Comptes Rendus. Mathématique PY - 2010 SP - 1087 EP - 1092 VL - 348 IS - 19-20 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2010.09.012/ DO - 10.1016/j.crma.2010.09.012 LA - en ID - CRMATH_2010__348_19-20_1087_0 ER -
%0 Journal Article %A Marica, Aurora %A Zuazua, Enrique %T Localized solutions and filtering mechanisms for the discontinuous Galerkin semi-discretizations of the $ 1-d$ wave equation %J Comptes Rendus. Mathématique %D 2010 %P 1087-1092 %V 348 %N 19-20 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2010.09.012/ %R 10.1016/j.crma.2010.09.012 %G en %F CRMATH_2010__348_19-20_1087_0
Marica, Aurora; Zuazua, Enrique. Localized solutions and filtering mechanisms for the discontinuous Galerkin semi-discretizations of the $ 1-d$ wave equation. Comptes Rendus. Mathématique, Tome 348 (2010) no. 19-20, pp. 1087-1092. doi : 10.1016/j.crma.2010.09.012. http://www.numdam.org/articles/10.1016/j.crma.2010.09.012/
[1] Dispersive behaviour of high order discontinuous Galerkin finite element method, Journal of Computational Physics, Volume 198 (2004) no. 1, pp. 106-130
[2] Discontinuous Galerkin approximation of the Laplace eigenproblem, Comput. Methods Appl. Mech. Engrg., Volume 195 (2006) no. 25–28, pp. 3483-3505
[3] Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., Volume 39 (2002), pp. 1749-1779
[4] S. Ervedoza, E. Zuazua, Propagation, observation and numerical approximation of waves, in preparation.
[5] Ensuring the well-posedness by analogy; Stokes problem and boundary control for the wave equation, Journal of Computational Physics, Volume 103 (1992) no. 2, pp. 189-221
[6] Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach, Encyclopedia of Mathematics and its Applications, vol. 117, Cambridge University Press, 2008
[7] Convergence of a multi-grid method for the control of waves, J. Eur. Math. Soc., Volume 11 (2009), pp. 351-391
[8] Numerical dispersive schemes for the nonlinear Schrödinger equation, SIAM. J. Numer. Anal., Volume 47 (2009) no. 2, pp. 1366-1390
[9] Localized solutions for the finite difference semi-discretization of the wave equation, C. R. Acad. Sci. Paris, Ser. I, Volume 348 (2010) no. 11–12, pp. 647-652
[10] Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures Appl., Volume 70 (1991), pp. 513-529
[11] Propagation, observation, control and numerical approximations of waves, SIAM Review, Volume 47 (2005) no. 2, pp. 197-243
Cité par Sources :
