On the Implementation of a Multiscale Hybrid High-Order Method

  • Matteo Cicuttin
  • Alexandre Ern
  • Simon Lemaire
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)


A multiscale Hybrid High-Order method has been introduced recently to approximate elliptic problems with oscillatory coefficients. In this work, with a view toward implementation, we describe the general workflow of the method and we present one possible way for accurately approximating the oscillatory basis functions by means of a monoscale Hybrid High-Order method deployed on a fine-scale mesh in each cell of the coarse-scale mesh.


  1. 1.
    A. Abdulle, W. E, B. Engquist, E. Vanden-Eijnden, The heterogeneous multiscale method. Acta Numer. 21, 1–87 (2012)Google Scholar
  2. 2.
    G. Allaire, R. Brizzi, A multiscale finite element method for numerical homogenization. SIAM Multiscale Model. Simul. 4(3), 790–812 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    M. Cicuttin, A. Ern, S. Lemaire, A hybrid high-order method for highly oscillatory elliptic problems. Comput. Methods Appl. Math. (2018). Scholar
  4. 4.
    M. Cicuttin, D.A. Di Pietro, A. Ern, Implementation of discontinuous skeletal methods on arbitrary-dimensional, polytopal meshes using generic programming. J. Comput. Appl. Math. 344, 852–874 (2018). MathSciNetCrossRefGoogle Scholar
  5. 5.
    B. Cockburn, D.A. Di Pietro, A. Ern, Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM: Math. Model. Numer. Anal. (M2AN) 50(3), 635–650 (2016)Google Scholar
  6. 6.
    D.A. Di Pietro, A. Ern, A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Eng. 283, 1–21 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D.A. Di Pietro, A. Ern, S. Lemaire, An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Methods Appl. Math. 14(4), 461–472 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Y. Efendiev, T.Y. Hou, Multiscale Finite Element Methods - Theory and Applications. Surveys and Tutorials in the Applied Mathematical Sciences, vol. 4 (Springer, New York, 2009)Google Scholar
  9. 9.
    Y. Efendiev, R. Lazarov, K. Shi, A multiscale HDG method for second order elliptic equations. Part I. Polynomial and homogenization-based multiscale spaces. SIAM J. Numer. Anal. 53(1), 342–369 (2015)zbMATHGoogle Scholar
  10. 10.
    C. Le Bris, F. Legoll, A. Lozinski, MsFEM à la Crouzeix–Raviart for highly oscillatory elliptic problems. Chin. Ann. Math. Ser. B 34(1), 113–138 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    L. Mu, J. Wang, X. Ye, A Weak Galerkin generalized multiscale finite element method. J. Comput. Appl. Math. 305, 68–81 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    D. Paredes, F. Valentin, H.M. Versieux, On the robustness of multiscale hybrid-mixed methods. Math. Comput. 86, 525–548 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Matteo Cicuttin
    • 1
    • 2
  • Alexandre Ern
    • 1
    • 2
  • Simon Lemaire
    • 3
    • 4
  1. 1.Université Paris-Est, CERMICS (ENPC)Marne-la-Vallée CedexFrance
  2. 2.Inria ParisParisFrance
  3. 3.École Polytechnique Fédérale de Lausanne, FSB-MATH-ANMCLausanneSwitzerland
  4. 4.Inria Lille - Nord EuropeVilleneuve d’AscqFrance

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