Skip to main content
SpringerLink
Log in

A Review of Hybrid High-Order Methods: Formulations, Computational Aspects, Comparison with Other Methods

  • Chapter
  • First Online:
Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 114))

Abstract

Hybrid High-Order (HHO) methods are formulated in terms of discrete unknowns attached to mesh faces and cells (hence, the term hybrid), and these unknowns are polynomials of arbitrary order k ≥ 0 (hence, the term high-order). HHO methods are devised from local reconstruction operators and a local stabilization term. The discrete problem is assembled cellwise, and cell-based unknowns can be eliminated locally by static condensation. HHO methods support general meshes, are locally conservative, and allow for a robust treatment of physical parameters in various situations, e.g., heterogeneous/anisotropic diffusion, quasi-incompressible linear elasticity, and advection-dominated transport. This paper reviews HHO methods for a variable-diffusion model problem with nonhomogeneous, mixed Dirichlet–Neumann boundary conditions, including both primal and mixed formulations. Links with other discretization methods from the literature are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J. Aghili, S. Boyaval, D.A. Di Pietro, Hybridization of mixed high-order methods on general meshes and application to the Stokes equations. Comput. Methods Appl. Math. 15 (2), 111–134 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. P.F. Antonietti, S. Giani, P. Houston, h p-version composite discontinuous Galerkin methods for elliptic problems on complicated domains. SIAM J. Sci. Comput. 35 (3), A1417–A1439 (2013)

    Google Scholar 

  3. R. Araya, C. Harder, D. Paredes, F. Valentin, Multiscale hybrid-mixed method. SIAM J. Numer. Anal. 51 (6), 3505–3531 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  5. D.N. Arnold, F. Brezzi, B. Cockburn, L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (5), 1749–1779 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. B. Ayuso de Dios, K. Lipnikov, G. Manzini, The nonconforming virtual element method. ESAIM: Math. Model Numer. Anal. (M2AN) 50 (3), 879–904 (2016)

    Google Scholar 

  7. F. Bassi, L. Botti, A. Colombo, D.A. Di Pietro, P. Tesini, On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations. J. Comput. Phys. 231 (1), 45–65 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Beirão da Veiga, F. Brezzi, L.D. Marini, Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51 (2), 794–812 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Beirão da Veiga, K. Lipnikov, G. Manzini, The Mimetic Finite Difference Method for Elliptic Problems. MS&A, vol. 11 (Springer, New York, 2014)

    Google Scholar 

  10. J. Bonelle, A. Ern, Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes. Math. Model. Numer. Anal. 48, 553–581 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Bonelle, A. Ern. Analysis of compatible discrete operator schemes for the Stokes equations on polyhedral meshes. IMA J. Numer. Anal. 35, 1672–1697 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Bonelle, D.A. Di Pietro, A. Ern, Low-order reconstruction operators on polyhedral meshes: application to compatible discrete operator schemes. Comput. Aided Geom. Des. 35–36, 27–41 (2015)

    Article  MathSciNet  Google Scholar 

  13. A. Bossavit, Computational electromagnetism and geometry. J. Jpn. Soc. Appl. Electromagn. Mech. 7–8, 150–159 (no. 1), 294–301 (no. 2), 401–408 (no. 3), 102–109 (no. 4), 203–209 (no. 5), 372–377 (no. 6) (1999–2000)

    Google Scholar 

  14. F. Brezzi, L.D. Marini, Virtual elements for plate bending problems. Comput. Methods Appl. Mech. Eng. 253, 455–462 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. F. Brezzi, G. Manzini, L.D. Marini, P. Pietra, A. Russo, Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differ. Equ. 16 (4), 365–378 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Brezzi, K. Lipnikov, M. Shashkov, Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43 (5), 1872–1896 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. F. Brezzi, K. Lipnikov, V. Simoncini, A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15 (10), 1533–1551 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Cangiani, E.H. Georgoulis, P. Houston, h p-Version discontinuous Galerkin methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 24 (10), 2009–2041 (2014)

    Google Scholar 

  19. P. Castillo, B. Cockburn, I. Perugia, D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38, 1676–1706 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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 

  21. B. Cockburn, J. Gopalakrishnan, R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second-order elliptic problems. SIAM J. Numer. Anal. 47 (2), 1319–1365 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. L. Codecasa, R. Specogna, F. Trevisan, A new set of basis functions for the discrete geometric approach. J. Comput. Phys. 19 (299), 7401–7410 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. D.A. Di Pietro, Cell-centered Galerkin methods for diffusive problems. Math. Model. Numer. Anal. 46 (1), 111–144 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. D.A. Di Pietro, J. Droniou. A hybrid high-order method for Leray-Lions elliptic equations on general meshes. Math. Comp. Accepted for publication. Preprint, arXiv:1508.01918 [math.NA]

    Google Scholar 

  25. D.A. Di Pietro, A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques & Applications, vol. 69 (Springer, Berlin/Heidelberg, 2012)

    Google Scholar 

  26. D.A. Di Pietro, A. Ern. Arbitrary-order mixed methods for heterogeneous anisotropic diffusion on general meshes. IMA J. Numer. Anal. (2016). Published online. doi:10.1093/imanum/drw003

    Google Scholar 

  27. D.A. Di Pietro, A. Ern, Equilibrated tractions for the Hybrid High-Order method. C. R. Acad. Sci. Paris Ser. I 353, 279–282 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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)

    Article  MathSciNet  Google Scholar 

  29. D.A. Di Pietro, A. Ern, Hybrid high-order methods for variable-diffusion problems on general meshes. C. R. Acad. Sci Paris Ser. I 353, 31–34 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. D.A. Di Pietro, J. Droniou, A. Ern, A discontinuous-skeletal method for advection-diffusionreaction on general meshes. SIAM J. Numer. Anal. 53 (5), 2135–2157 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. D.A. Di Pietro, S. Lemaire, An extension of the Crouzeix–Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow. Math. Comput. 84, 1–31 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. 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)

    Article  MathSciNet  MATH  Google Scholar 

  33. K. Domelevo, P. Omnes, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM Math. Model. Numer. Anal. 39 (6), 1203–1249 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. J. Droniou, R. Eymard, A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math. 105, 35–71 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Droniou, R. Eymard, T. Gallouët, R. Herbin, A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci. 20 (2), 1–31 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. J. Droniou, R. Eymard, T. Gallouët, R. Herbin, Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. Math. Models Methods Appl. Sci. 23, 2395–2432 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Ern, J.-L. Guermond, Discontinuous Galerkin methods for Friedrichs’ systems. I. General theory. SIAM J. Numer. Anal. 44 (2), 753–778 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. R. Eymard, T. Gallouët, R. Herbin, Finite volume methods, in Techniques of Scientific Computing (Part III), ed. by P.G. Ciarlet, J.-L. Lions. Handbook of Numerical Analysis, vol. 7 (North-Holland, Amsterdam, 2000), pp. 713–1020

    Google Scholar 

  39. R. Eymard, T. Gallouët, R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes. SUSHI: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30 (4), 1009–1043 (2010)

    MATH  Google Scholar 

  40. C. Harder, D. Paredes, F. Valentin, A family of multiscale hybrid-mixed finite element methods for the Darcy equation with rough coefficients. J. Comput. Phys. 245, 107–130 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  41. R. Herbin, F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, in Finite Volumes for Complex Applications V, ed. by R. Eymard, J.-M. Hérard (Wiley, London, 2008), pp. 659–692

    Google Scholar 

  42. C. Lehrenfeld, Hybrid discontinuous Galerkin methods for solving incompressible flow problems. Ph.D. thesis, Rheinisch-Westfälischen Technischen Hochschule Aachen, 2010

    Google Scholar 

  43. K. Lipnikov, G. Manzini, A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation. J. Comput. Phys. 272, 360–385 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  44. E. Tonti, On the formal structure of physical theories. Quaderni dei Gruppi di Ricerca Matematica del CNR (1975)

    Google Scholar 

  45. J. Wang, X. Ye, A weak Galerkin element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103–115 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  46. J. Wang, X. Ye, A weak Galerkin mixed finite element method for second-order elliptic problems. Math. Comput. 83 (289), 2101–2126 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Montpelliérain Alexander Grothendieck, University of Montpellier, 34095, Montpellier, France

    Daniele A. Di Pietro

  2. University Paris-Est, CERMICS (ENPC), 6–8 Avenue Blaise Pascal, 77455, Marne-la-Vallée Cedex 2, France

    Alexandre Ern & Simon Lemaire

Authors
  1. Daniele A. Di Pietro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexandre Ern
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Simon Lemaire
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexandre Ern .

Editor information

Editors and Affiliations

  1. Department of Mathematics & Statistics, University of Strathclyde, Glasgow, United Kingdom

    Gabriel R. Barrenechea

  2. Istituto di Matematica Applicata e Tecnologie Informatiche - Pavia, Consiglio Nazionale delle Ricerche, Pavia, Italy

    Franco Brezzi

  3. Department of Mathematics, University of Leicester, Leicester, United Kingdom

    Andrea Cangiani

  4. Department of Mathematics, National Technical University of Athens, Zografou, Greece

    Emmanuil H. Georgoulis

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Di Pietro, D.A., Ern, A., Lemaire, S. (2016). A Review of Hybrid High-Order Methods: Formulations, Computational Aspects, Comparison with Other Methods. In: Barrenechea, G., Brezzi, F., Cangiani, A., Georgoulis, E. (eds) Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 114. Springer, Cham. https://doi.org/10.1007/978-3-319-41640-3_7

Download citation

Publish with us

Policies and ethics

Search

Navigation