Abstract
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.
Similar content being viewed by others
References
Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6(3–4), 405–432 (2002). https://doi.org/10.1023/A:10212911
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). https://doi.org/10.1137/S1064827595293582
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). https://doi.org/10.1002/num.20320
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). https://doi.org/10.1051/m2an/2010021
Ayuso de Dios, B., Lipnikov, K., Manzini, G.: The nonconforming virtual element method. ESAIM Math. Model. Numer. Anal. 50(3), 879–904 (2016). https://doi.org/10.1051/m2an./2015090
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). https://doi.org/10.1142/S0218202512500492
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). https://doi.org/10.1142/S021820251440003X
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). https://doi.org/10.1142/S0218202516500160
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). https://doi.org/10.1007/978-3-319-02663-3
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). https://doi.org/10.1051/m2an/2017036
Brenner, S.C., Guan, Q., Sung, L.-Y.: Some estimates for virtual element methods. Comput. Methods Appl. Math. 17(4), 553–574 (2017). https://doi.org/10.1515/cmam-2017-0008
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). https://doi.org/10.1007/978-0-387-75934-0. ISBN: 978-0-387-75933-3
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). https://doi.org/10.1093/imanum/drw036
Chatzipantelidis, P.: Finite volume methods for elliptic PDE’s: a newapproach. M2AN Math. Model. Numer. Anal. 36(2), 307–324 (2002). https://doi.org/10.1051/m2an:2002014
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). https://doi.org/10.1090/S0025-5718-99-01192-8
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: 0-89871-514-8
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). https://doi.org/10.1051/m2an/2015051
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). https://doi.org/10.1142/S0218202517500191
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). https://doi.org/10.1137/140993971
Di Pietro, D.A., Droniou, J., Manzini, G.: Discontinuous skeletal gradient discretisation methods on polytopalmeshes. J. Comput. Phys. 355, 397–425 (2018). https://doi.org/10.1016/j.jcp.2017.11.018
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). https://doi.org/10.1093/imanum/drw003
Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques and Applications, vol. 69. Springer, Berlin (2012). https://doi.org/10.1007/978-3-642-22980-0
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). https://doi.org/10.1007/978-3-319-41640-3
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: 978-3-319-94675
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). https://doi.org/10.1016/j.crma.2014.10.013
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). https://doi.org/10.1142/S0218202514400041
Droniou, J., Eymard, R.: A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math. 105, 35–71 (2006). https://doi.org/10.1007/s00211-006-0034-1
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)
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). https://doi.org/10.1007/978-3-319-79042-8. https://hal.archives-ouvertes.fr/hal-01382358
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). https://doi.org/10.1142/S0218202510004222
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). https://doi.org/10.1093/imanum/drx028. arxiv: 1602.07359
Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34(150), 441–463 (1980)
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)
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). https://doi.org/10.1515/cmam-2017-0058
Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, New York (2004)
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). https://doi.org/10.1002/(SICI)1098-2426(200005)16:3%3c285::AID-NUM2%3e3.0.CO;2-3
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). https://doi.org/10.1093/imanum/drn084
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)
Gudi, T.: A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comput. 79(272), 2169–2189 (2010). https://doi.org/10.1090/S0025-5718-10-02360-4
Lipnikov, K., Manzini, G.: A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation. J. Comput. Phys. 272, 360–385 (2014). https://doi.org/10.1016/j.jcp.2014.04.021
Mishev, I.D.: Finite volume element methods for non-definite problems. Numer. Math. 83(1), 161–175 (1999). https://doi.org/10.1007/s002110050443
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)
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)
Strang, G., Fix, G.: An Analysis of the Finite Element Method, 2nd edn, p. x+402. Wellesley-Cambridge Press, Wellesley (2008)
Tartar, L.: Personal Communication. Dec. 26 (2015)
Wang, J., Ye, X.: A weak Galerkin element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103–115 (2013). https://doi.org/10.1016/j.cam.2012.10.003
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Di Pietro, D.A., Droniou, J. A third Strang lemma and an Aubin–Nitsche trick for schemes in fully discrete formulation. Calcolo 55, 40 (2018). https://doi.org/10.1007/s10092-018-0282-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10092-018-0282-3
Keywords
- Strang lemma
- Consistency
- Error estimate
- Aubin–Nitsche trick
- Virtual element methods
- Finite volume methods
- Oblique elliptic projector