Comptes Rendus
Numerical analysis
Finite-element heterogeneous multiscale method for the Helmholtz equation
Comptes Rendus. Mathématique, Volume 352 (2014) no. 9, pp. 755-760.

We show that the standard Finite Element Heterogeneous Multiscale Method (FE-HMM) can be used to approximate the effective behavior of solutions to the classical Helmholtz equation in highly oscillatory media. Using a novel combination of well-known results about FE-HMM and the notion of T-coercivity, we derive an a priori error bound. Numerical experiments corroborate the analytical findings.

Nous montrons que la méthode multi-échelle hétérogène d'éléments finis (FE-HMM) peut être utilisée pour approcher le comportement effectif des solutions de l'équation de Helmholtz classique dans des milieux rapidement oscillants. À l'aide de cette méthode et de la notion de T-coercivité, nous établissons une borne a priori de l'erreur. Des expériences numériques corroborent les résultats théoriques.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2014.07.006
Patrick Ciarlet 1; Christian Stohrer 1

1 Laboratoire POEMS, UMA, ENSTA ParisTech, 828, boulevard des Maréchaux, 91762 Palaiseau cedex, France
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Patrick Ciarlet; Christian Stohrer. Finite-element heterogeneous multiscale method for the Helmholtz equation. Comptes Rendus. Mathématique, Volume 352 (2014) no. 9, pp. 755-760. doi : 10.1016/j.crma.2014.07.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.07.006/

[1] A. Abdulle The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs, GAKUTO Int. Ser. Math. Sci. Appl., Volume 31 (2009), pp. 133-181

[2] A. Abdulle; W. E; B. Engquist; E. Vanden-Eijnden The heterogeneous multiscale method, Acta Numer., Volume 21 (2012), pp. 1-87

[3] G. Allaire Homogenization and two-scale convergence, SIAM J. Math. Anal., Volume 23 (1992) no. 6, pp. 1482-1518

[4] P. Ciarlet T-coercivity: application to the discretization of Helmholtz-like problems, Comput. Math. Appl., Volume 64 (2012) no. 1, pp. 22-34

[5] P.G. Ciarlet The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, vol. 40, SIAM, 2002 (reprinted from the 1978 original)

[6] W. E; B. Engquist The heterogeneous multiscale methods, Commun. Math. Sci., Volume 1 (2003) no. 1, pp. 87-132

[7] O.A. Oleinik; A.S. Shamaev; G.A. Yosifian Mathematical Problems in Elasticity and Homogenization, Studies in Mathematics and Applications, vol. 26, Elsevier, 1992

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