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AIMS Mathematics

2017, Volume 2, Issue 3: 458-478. doi: 10.3934/Math.2017.2.458
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Research article

Localized Orthogonal Decomposition for two-scale Helmholtz-typeproblems

  • Applied Mathematics, University of Münster, D-48149 Münster, Germany
  • Received: 05 May 2017 Accepted: 14 August 2017 Published: 14 August 2017

  • In this paper, we present a Localized Orthogonal Decomposition (LOD) in Petrov-Galerkin formulation for a two-scale Helmholtz-type problem. The two-scale problem is, for instance, motivated from the homogenization of the Helmholtz equation with high contrast, studied together with a corresponding multiscale method in a previous paper of the authors. There, an unavoidable resolution condition on the mesh sizes in terms of the wave number has been observed, which is known as "pollution e ect" in the finite element literature. Following ideas of Gallistl and Peterseim, we use standard finite element functions for the trial space, whereas the test functions are enriched by solutions of subscsale problems (solved on a finer grid) on local patches. Provided that the oversampling parameter m, which indicates the size of the patches, is coupled logarithmically to the wave number, we obtain a quasi-optimal method under a reasonable resolution of a few degrees of freedom per wave length, thus overcoming the pollution effect. In the two-scale setting, the main challenges for the LOD lie in the coupling of the function spaces and in the periodic boundary conditions.

    Citation: Mario Ohlberger, Barbara Verfürth. Localized Orthogonal Decomposition for two-scale Helmholtz-typeproblems[J]. AIMS Mathematics, 2017, 2(3): 458-478. doi: 10.3934/Math.2017.2.458

    Related Papers:

  • In this paper, we present a Localized Orthogonal Decomposition (LOD) in Petrov-Galerkin formulation for a two-scale Helmholtz-type problem. The two-scale problem is, for instance, motivated from the homogenization of the Helmholtz equation with high contrast, studied together with a corresponding multiscale method in a previous paper of the authors. There, an unavoidable resolution condition on the mesh sizes in terms of the wave number has been observed, which is known as "pollution e ect" in the finite element literature. Following ideas of Gallistl and Peterseim, we use standard finite element functions for the trial space, whereas the test functions are enriched by solutions of subscsale problems (solved on a finer grid) on local patches. Provided that the oversampling parameter m, which indicates the size of the patches, is coupled logarithmically to the wave number, we obtain a quasi-optimal method under a reasonable resolution of a few degrees of freedom per wave length, thus overcoming the pollution effect. In the two-scale setting, the main challenges for the LOD lie in the coupling of the function spaces and in the periodic boundary conditions.


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    [1] A. Abdulle and P. Henning, Localized orthogonal decomposition method for the wave equation with a continuum of scales, Math. Comp., 86 (2017), 549-587.
    [2] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
    [3] I. M. Babuška and S. A. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM Rev., 42 (2000), 451-484.
    [4] G. Bouchitté, C. Bourel, and D. Felbacq, Homogenization of the 3D Maxwell system near resonances and artificial magnetism, C. R. Math. Acad. Sci. Paris, 347 (2009), 571-576.
    [5] G. Bouchitté and D. Felbacq, Homogenization near resonances and artificial magnetism from dielectrics, C. R. Math. Acad. Sci. Paris, 339 (2004), 377-382.
    [6] G. Bouchitté and B. Schweizer, Homogenization of Maxwell's equations in a split ring geometry, Multiscale Model. Simul., 8 (2010), 717-750.
    [7] G. Bouchitté and B. Schweizer, Plasmonic waves allow perfect transmission through subwavelength metallic gratings, Netw. Heterog. Media, 8 (2013), 857-878.
    [8] D. L. Brown, D. Gallistl, and D. Peterseim, Multiscale Petrov-Galerkin method for highfrequency heterogeneous Helmholtz equations, In M. Griebel and M. A. Schweitzer, editors, Meshfree Methods for Partial Differential Equations Ⅵ, Lecture Notes in Computational Science and Engineering, 2016.
    [9] H. Chen, P. Lu, and X. Xu, A hybridizable discontinuous Galerkin method for the Hlmholtz equation with high wave number, SIAM J. Numer. Anal., 51 (2013), 2166-2188.
    [10] K. Cherednichenko and S. Cooper, Homogenization of the system of high-contrast Maxwell equations, Mathematika, 61 (2015), 475-500.
    [11] A. Efros and A. Pokrovsky, Dielectric photonic crystal as medium with negative electric permittivity and magnetic permeability, Solid State Communications, 129 (2004), 643-647.
    [12] D. Elfverson, V. Ginting, and P. Henning, On multiscale methods in Petrov-Galerkin formulation, Numer. Math., 131 (2015), 643-682.
    [13] C. Engwer, P. Henning, A. Målqvist, and D. Peterseim, Effcient implementation of the localized orthogonal decomposition method, arXiv.
    [14] S. Esterhazy and J. M. Melenk, On stability of discretizations of the Helmholtz equation, Numerical analysis of multiscale problems of Lect. Notes Comput. Sci. Eng., Springer, Heidelberg, 83 (2012), 285-324,
    [15] D. Felbacq and G. Bouchitté, Homogenization of a set of parallel fibres, Waves Random Media, 7 (1997), 245-256.
    [16] D. Gallistl and D. Peterseim, Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering, Comput. Methods Appl. Mech. Engrg., 295 (2015), 1-17.
    [17] R. Griesmaier and P. Monk, Error analysis for a hybridizable discontinuous Galerkin method for the Helmholtz equation, J. Sci. Comput., 49 (2011), 291-310.
    [18] F. Hellman, P. Henning, and A. Målqvist, Multiscale mixed finite elements, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 12691298.
    [19] P. Henning, P. Morgenstern, and D. Peterseim, Multiscale partition of unity of Lecture Notes in Computational Science and Engineering, chapter Meshfree Methods for Partial Differential Equations Ⅶ, Springer International Publishing, 100 (2014), 185-204.
    [20] P. Henning, M.Ohlberger, and B.Verfürth, A new Heterogeneous Multiscale Method for timeharmonic Maxwell's equations, SIAM J. Numer. Anal., 54 (2016), 3493-3522.
    [21] P. Henning and A. Persson, A multiscale method for linear elasticity reducing Poisson locking, Comput. Methods Appl. Mech. Engrg., 310 (2016), 156171.
    [22] P. Henning and A. Målqvist, Localized orthogonal decomposition techniques for boundary value problems, SIAM J. Sci. Comput., 36 (2014), A1609-A1634.
    [23] R. Hiptmair, A. Moiola, and I. Perugia, A Survey of Trefftz methods for the Helmholtz Equation, Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations (eds. G. R. Barrenechea, A. Cangiani and E. H. Geogoulis), Lecture Notes in Computational Science and Engineering (LNCSE), Springer, Accepted for publication; Preprint arXiv: 1505. 04521.
    [24] R. Hiptmair, A. Moiola, and I. Perugia, Plane Wave Discontinuous Galerkin Methods: Exponential Convergence of the hp-version, Found. Comp. Math., 16 (2016), 637675.
    [25] T. J. R. Hughes and G. Sangalli, Variational multiscale analysis: the fine-scale Green's function, projection, optimization, localization, and stabilized methods, SIAM J. Numer. Anal., 45 (2007), 539-557.
    [26] T. J. R. Hughes, G. R. Feijóo, L. Mazzei, and J. B. Quincy, The variational multiscale method-a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg., 166 (1998), 3-24.
    [27] P. C. Jr. and C. Stohrer, Finite Element Heterogeneous Multiscale Method for the classical Helmholtz Equation, C. R. Acad. Sci., 352 (2014), 755-760.
    [28] A. Lamacz and B. Schweizer, A negative index meta-material for Maxwell's equations, SIAM J. Math. Anal., 48 (2016), 41554174.
    [29] C. Luo, S. G. Johnson, J. Joannopolous, and J. Pendry, All-angle negative refraction without negative effective index, Phys. Rev. B, 65 (2011).
    [30] A. Målqvist and D. Peterseim, Localization of elliptic multiscale problems, Math. Comp., 83 (2014), 2583-2603.
    [31] A. Målqvist and D. Peterseim, Computation of eigenvalues by numerical upscaling, Numer. Math., 130 (2015), 337-361.
    [32] J. M. Melenk and S. Sauter, Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation, SIAM J. Numer. Anal., 49 (2011), 1210-1243.
    [33] S. O'Brien and J. B. Pendry, Photonic band-gap effects and magnetic activity in dielectric composites, Journal of Physics: Condensed Matter, 14 (2002), 4035.
    [34] M. Ohlberger and B. Verfürth, A new Heterogenous Multiscale Method for the Helmholtz equation with high contrast, arXiv, 2016.
    [35] M. Ohlberger, A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems, Multiscale Model. Simul., 4 (2005), 88-114.
    [36] I. Perugia, P. Pietra, and A. Russo, A Plane Wave Virtual Element Method for the Helmholtz Problem, ESAIM Math. Model. Numer. Anal., 50 (2016), 783-808.
    [37] D. Peterseim, Eliminating the pollution effect in Helmholtz problems by local subscale correction, Math. Comp., 86 (2017), 1005-1036.
    [38] D. Peterseim, Variational multiscale stabilization and the exponential decay of fine-scale correctors, Springer, 114 (2016).
    [39] A. Pokrovsky and A. Efros, Diffraction theory and focusing of light by a slab of left-handed material, Physica B: Condensed Matter, 338 (2003), 333-337,
    [40] S. A. Sauter, A refined finite element convergence theory for highly indefinite Helmholtz problems, Computing, 78 (2006), 101-115.
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