, 55:40 | Cite as

A third Strang lemma and an Aubin–Nitsche trick for schemes in fully discrete formulation

  • Daniele A. Di Pietro
  • Jérôme DroniouEmail author


In this work, we present an abstract error analysis framework for the approximation of linear partial differential equation problems in weak formulation. We consider approximation methods in fully discrete formulation, where the discrete and continuous spaces are possibly not embedded in a common space. A proper notion of consistency is designed, and, under a classical inf–sup condition, it is shown to bound the approximation error. This error estimate result is in the spirit of Strang’s first and second lemmas, but applicable in situations not covered by these lemmas (because of a fully discrete approximation space). An improved estimate is also established in a weaker norm, using the Aubin–Nitsche trick. We then apply these abstract estimates to an anisotropic heterogeneous diffusion model and two classical families of schemes for this model: virtual element and finite volume methods. For each of these methods, we show that the abstract results yield new error estimates with a precise and mild dependency on the local anisotropy ratio. A key intermediate step to derive such estimates for virtual element methods is proving optimal approximation properties of the oblique elliptic projector in weighted Sobolev seminorms. This is a result whose interest goes beyond the specific model and methods considered here. We also obtain, to our knowledge, the first clear notion of consistency for finite volume methods, which leads to a generic error estimate involving the fluxes and valid for a wide range of finite volume schemes. An important application is the first error estimate for multi-point flux approximation L and G methods.


Strang lemma Consistency Error estimate Aubin–Nitsche trick Virtual element methods Finite volume methods Oblique elliptic projector 

Mathematics Subject Classification

65N08 65N12 65N15 65N30 



The work of the first author was supported by Agence Nationale de la Recherche Grants HHOMM (ANR-15-CE40-0005) and fast4hho (ANR-17-CE23-0019). The work of the second author was partially supported by the Australian Government through the Australian Research Council’s Discovery Projects funding scheme (Project Number DP170100605). Fruitful discussions with Simon Lemaire (INRIA Lille - Nord Europe) are gratefully acknowledged.


  1. 1.
    Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6(3–4), 405–432 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aavatsmark, I., Barkve, T., Bøe, O., Mannseth, T.: Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods. SIAM J. Sci. Comput. 19(5), 1700–1716 (1998). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aavatsmark, I., Eigestad, G.T., Mallison, B.T., Nordbotten, J.M.: A compact multipoint flux approximation method with improved robustness. Numer. Methods Partial Differ. Equ. 24(5), 1329–1360 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Agélas, L., Di Pietro, D.A., Droniou, J.: The G method for heterogeneous anisotropic diffusion on general meshes. ESAIM Math. Model. Numer. Anal. 44(4), 597–625 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ayuso de Dios, B., Lipnikov, K., Manzini, G.: The nonconforming virtual element method. ESAIM Math. Model. Numer. Anal. 50(3), 879–904 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. (M3AS) 199(23), 199–214 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Beirão da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: The hitchhiker’s guide to the virtual element method. Math. Models Methods Appl. Sci. 24(8), 1541–1573 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Beirão da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: Virtual element method for general second order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26(4), 729–750 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Beirão da Veiga, L., Lipnikov, K., Manzini, G.: The Mimetic Finite Difference Method for Elliptic Problems. Modeling, Simulation and Applications, vol. 11. Springer, Berlin (2014). CrossRefzbMATHGoogle Scholar
  10. 10.
    Boffi, D., Di Pietro, D.A.: Unified formulation and analysis of mixed and primal discontinuous skeletal methods on polytopal meshes. ESAIM Math. Model. Numer. Anal. 52(1), 1–28 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Brenner, S.C., Guan, Q., Sung, L.-Y.: Some estimates for virtual element methods. Comput. Methods Appl. Math. 17(4), 553–574 (2017). MathSciNetCrossRefGoogle Scholar
  12. 12.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15, 3rd edn, p. xviii++397. Springer, New York (2008). ISBN: 978-0-387-75933-3CrossRefGoogle Scholar
  13. 13.
    Cangiani, A., Manzini, G., Sutton, O.J.: Conforming and nonconforming virtual element methods for elliptic problems. IMA J. Numer. Anal. 37(3), 1317–1354 (2017). MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chatzipantelidis, P.: Finite volume methods for elliptic PDE’s: a newapproach. M2AN Math. Model. Numer. Anal. 36(2), 307–324 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chou, S.-H., Li, Q.: Error estimates in \(L^2\), \(H^{1}\) and \(L^{\infty }\) in covolume methods for elliptic and parabolic problems: a unified approach. Math. Comput. 69(229), 103–120 (2000). CrossRefzbMATHGoogle Scholar
  16. 16.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, vol. 40. Reprint of the 1978 Original [North-Holland, Amsterdam; MR0520174 (58 #25001)], p. xxviii+530. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). ISBN: 0-89871-514-8Google Scholar
  17. 17.
    Cockburn, B., Di Pietro, D.A., Ern, A.: Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM Math. Model. Numer. Anal. 50(3), 635–650 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Di Pietro, D.A., Droniou, J.: A hybrid high-order method for Leray–Lions elliptic equations on general meshes. Math. Comput. 86(307), 2159–2191 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Di Pietro, D.A., Droniou, J., Ern, A.: A discontinuous-skeletal method for advection–diffusion–reaction on general meshes. SIAM J. Numer. Anal. 53(5), 2135–2157 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Di Pietro, D.A., Droniou, J., Manzini, G.: Discontinuous skeletal gradient discretisation methods on polytopalmeshes. J. Comput. Phys. 355, 397–425 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Di Pietro, D.A., Ern, A.: Arbitrary-order mixed methods for heterogeneous anisotropic diffusion on general meshes. IMA J. Numer. Anal. 37(1), 40–63 (2017). MathSciNetCrossRefGoogle Scholar
  22. 22.
    Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques and Applications, vol. 69. Springer, Berlin (2012). CrossRefzbMATHGoogle Scholar
  23. 23.
    Di Pietro, D.A., Ern, A., Lemaire, S.: 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). MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Di Pietro, D. A., Tittarelli, R.: An introduction to Hybrid High-Order methods. In: Di Pietro, D. A., Ern, A., Formaggia, L. (eds.) Numerical Methods for PDEs: State of the Art Techniques. Springer (2018). ISBN: 978-3-319-94675Google Scholar
  25. 25.
    Di Pietro, D.A., Ern, A.: Hybrid high-order methods for variable-diffusion problems on general meshes. C. R. Math. Acad. Sci. Paris 353(1), 31–34 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Droniou, J.: Finite volume schemes for diffusion equations: introduction to and review of modern methods. Math. Models Methods Appl. Sci. 24(8), 1575–1619 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Droniou, J., Eymard, R.: A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math. 105, 35–71 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Droniou, J., Eymard, R.: The asymmetric gradient discretisation method. In: Finite Volumes for Complex Applications VIII-Methods and Theoretical Aspects, vol. 199. Springer Proceedings in Mathematics and Statistics. Springer, Cham, pp. 311–319 (2017)Google Scholar
  29. 29.
    Droniou, J., Eymard, R., Gallouët, T., Guichard, C., Herbin, R.: The Gradient Discretisation Method. Mathematics and Applications, vol. 82. Springer, p. 511 (2018). ISBN: 978-3-319-79041-1 (Softcover) 978- 3-319-79042-8 (eBook). CrossRefGoogle Scholar
  30. 30.
    Droniou, J., Eymard, R., Gallouët, T., Herbin, R.: A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci. (M3AS) 20(2), 1–31 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Droniou, J., Nataraj, N.: Improved L2 estimate for gradient schemes and super-convergence of the TPFAfinite volume scheme. IMA J. Numer. Anal. 38(3), 1254–1293 (2018). arxiv: 1602.07359 MathSciNetCrossRefGoogle Scholar
  32. 32.
    Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34(150), 441–463 (1980)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Edwards, M.G., Rogers, C.F.: A flux continuous scheme for the full tensor pressure equation. In: Proceedings of the 4th European Conference on the Mathematics of Oil Recovery, Vol. D. Røros, Norway (1994)Google Scholar
  34. 34.
    Ern, A., Guermond, J.-L.: Abstract nonconforming error estimates and application to boundary penalty methods for diffusion equations and time-harmonic Maxwell’s equations. Comput. Methods Appl. Math. (2018). MathSciNetCrossRefGoogle Scholar
  35. 35.
    Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, New York (2004)CrossRefGoogle Scholar
  36. 36.
    Ewing, R., Lazarov, R., Lin, Y.: Finite volume element approximations of nonlocal reactive flows in porous media. Numer. Methods Partial Differ. Equ. 16(3), 285–311 (2000).;2-3 MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Eymard, R., Gallouët, T., Herbin, R.: 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). MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Ciarlet, P.G., Lions, J.-L. (eds.) Handbook of Numerical Analysis, VII. Techniques of Scientific Computing, Part III, pp. 713–1020. North- Holland, Amsterdam (2000)Google Scholar
  39. 39.
    Gudi, T.: A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comput. 79(272), 2169–2189 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Lipnikov, K., Manzini, G.: A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation. J. Comput. Phys. 272, 360–385 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Mishev, I.D.: Finite volume element methods for non-definite problems. Numer. Math. 83(1), 161–175 (1999). MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Stampacchia, G.: Le probléme de Dirichlet pour les èquations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15(fasc. 1), 189–258 (1965)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Strang, G.: Variational crimes in the finite element method. In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, pp. 689–710 (Proceedings of Symposia, University Maryland, Baltimore, MD, 1972). Academic Press, New York (1972)CrossRefGoogle Scholar
  44. 44.
    Strang, G., Fix, G.: An Analysis of the Finite Element Method, 2nd edn, p. x+402. Wellesley-Cambridge Press, Wellesley (2008)zbMATHGoogle Scholar
  45. 45.
    Tartar, L.: Personal Communication. Dec. 26 (2015)Google Scholar
  46. 46.
    Wang, J., Ye, X.: A weak Galerkin element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103–115 (2013). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica del Consiglio Nazionale delle Ricerche 2018

Authors and Affiliations

  1. 1.Institut Montpelliérain Alexander GrothendieckUniv. Montpellier, CNRSMontpellierFrance
  2. 2.School of Mathematical SciencesMonash UniversityMelbourneAustralia

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