Localized solutions and filtering mechanisms for the discontinuous Galerkin semi-discretizations of the $ 1-d$ wave equation

Aurora Marica; Enrique Zuazua

doi:10.1016/j.crma.2010.09.012

Partial Differential Equations/Numerical Analysis

Localized solutions and filtering mechanisms for the discontinuous Galerkin semi-discretizations of the

1 - d

wave equation
[Solutions localisées et mécanismes de filtrage pour les approximations de Galerkin discontinues de l'équation des ondes]

Présenté par : Roland Glowinski

Aurora Marica ¹ ; Enrique Zuazua ^1,²

¹ BCAM – Basque Center for Applied Mathematics, Bizkaia Technology Park 500, 48160 Derio, Basque Country, Spain
² Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011 Bilbao, Basque Country, Spain

Comptes Rendus. Mathématique, Volume 348 (2010) no. 19-20, pp. 1087-1092.

Résumé
Abstract

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) $P_{1}$ 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 $1 - d$ wave equation obtained through a $P_{1}$ 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.

Reçu le : 2010-04-13
Accepté le : 2010-09-10
Publié le : 2010-09-23

DOI : 10.1016/j.crma.2010.09.012

Affiliations des auteurs :

Aurora Marica ¹ ; Enrique Zuazua ^{1, 2}

¹ BCAM – Basque Center for Applied Mathematics, Bizkaia Technology Park 500, 48160 Derio, Basque Country, Spain
² Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011 Bilbao, Basque Country, Spain

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     author = {Aurora Marica and Enrique Zuazua},
     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},
}

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Aurora Marica; Enrique Zuazua. Localized solutions and filtering mechanisms for the discontinuous Galerkin semi-discretizations of the $ 1-d$ wave equation. Comptes Rendus. Mathématique, Volume 348 (2010) no. 19-20, pp. 1087-1092. doi : 10.1016/j.crma.2010.09.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.09.012/

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