Computational and Applied Mathematics

, Volume 37, Issue 4, pp 4023–4054 | Cite as

Discontinuous Galerkin gradient discretisations for the approximation of second-order differential operators in divergence form

  • Robert EymardEmail author
  • Cindy Guichard


We include in the gradient discretisation method (GDM) framework two numerical schemes based on discontinuous Galerkin approximations: the symmetric interior penalty Galerkin (SIPG) method, and the scheme obtained by averaging the jumps in the SIPG method. We prove that these schemes meet the main mathematical gradient discretisation properties on any kind of polytopal mesh, by adapting discrete functional analysis properties to our precise geometrical hypotheses. Therefore, these schemes inherit the general convergence properties of the GDM, which hold for instance in the cases of the p-Laplace problem and of the anisotropic and heterogeneous diffusion problem. This is illustrated by simple 1D and 2D numerical examples.


Gradient discretisation method Discontinuous Galerkin method Symmetric interior penalty Galerkin scheme Discrete functional analysis Polytopal meshes 

Mathematics Subject Classification



  1. Arnold DN (1982) An interior penalty finite element method with discontinuous elements. SIAM J Numer Anal 19(4):742–760MathSciNetCrossRefzbMATHGoogle Scholar
  2. Arnold DN, Brezzi F, Cockburn B, Marini LD (2001/2002) Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J Numer Anal 39(5):1749–1779Google Scholar
  3. Brenner SC, Owens L (2007) A weakly over-penalized non-symmetric interior penalty method. JNAIAM J Numer Anal Ind Appl Math 2(1–2):35–48MathSciNetzbMATHGoogle Scholar
  4. Burman E, Ern A (2008) Discontinuous Galerkin approximation with discrete variational principle for the nonlinear Laplacian. C R Math Acad Sci Paris 346(17–18):1013–1016MathSciNetCrossRefzbMATHGoogle Scholar
  5. Burman E, Zunino P (2006) A domain decomposition method for partial differential equations with non-negative form based on interior penalties. SIAM J Numer Anal 44(4):1612–1638MathSciNetCrossRefzbMATHGoogle Scholar
  6. Di Pietro DA, Droniou J (2017) A hybrid high-order method for Leray–Lions elliptic equations on general meshes. Math Comput 86(307):2159–2191MathSciNetCrossRefzbMATHGoogle Scholar
  7. Di Pietro DA, Ern A (2010) Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier–Stokes equations. Math Comput 79(271):1303–1330MathSciNetCrossRefzbMATHGoogle Scholar
  8. Di Pietro DA, Ern A (2012) Mathematical aspects of discontinuous Galerkin methods, vol 69. Mathématiques & applications (Berlin) [Mathematics & applications]. Springer, HeidelbergCrossRefzbMATHGoogle Scholar
  9. Droniou J, Eymard R, Gallouët T, Guichard C, Herbin R (2016) The gradient discretisation method. working paper or preprint.
  10. Epshteyn Y, Rivière B (2007) Estimation of penalty parameters for symmetric interior penalty Galerkin methods. J Comput Appl Math 206(2):843–872MathSciNetCrossRefzbMATHGoogle Scholar
  11. Ern A, Stephansen AF, Zunino P (2008) A discontinuous Galerkin method with weighted averages for advection–diffusion equations with locally small and anisotropic diffusivity. IMA J Numer Anal 29(2):235–256MathSciNetCrossRefzbMATHGoogle Scholar
  12. Eymard R, Guichard C, Herbin R (2012) Small-stencil 3d schemes for diffusive flows in porous media. M2AN 46:265–290MathSciNetCrossRefzbMATHGoogle Scholar
  13. Hansbo P, Larson MG (2003) Discontinuous Galerkin and the Crouzeix–Raviart element: application to elasticity. M2AN Math Model Numer Anal 37(1):63–72MathSciNetCrossRefzbMATHGoogle Scholar
  14. Herbin R, Hubert F (2008) Benchmark on discretization schemes for anisotropic diffusion problems on general grids. In: Finite volumes for complex applications V. ISTE, London, pp 659–692Google Scholar
  15. John L, Neilan M, Smears I (2016) Stable discontinuous Galerkin FEM without penalty parameters. In: Numerical mathematics and advanced applications ENUMATH 2015. Springer, pp 165–173Google Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017

Authors and Affiliations

  1. 1.Laboratoire d’Analyse et de Mathématiques Appliquées, UPEC, UPEM, UMR8050 CNRSUniversité Paris-Est Marne-la-ValléeMarne-la-ValléeFrance
  2. 2.Laboratoire Jacques-Louis Lions, UMR CNRS 7598Sorbonne Universités, UPMC Univ. Paris 6ParisFrance
  3. 3.ANGE Project-Team (Inria, Cerema, UPMC, CNRS)ParisFrance

Personalised recommendations