Skip to main content Accessibility help

Cell centered Galerkin methods for diffusive problems

  • Daniele A. Di Pietro (a1)


In this work we introduce a new class of lowest order methods for diffusive problems on general meshes with only one unknown per element. The underlying idea is to construct an incomplete piecewise affine polynomial space with optimal approximation properties starting from values at cell centers. To do so we borrow ideas from multi-point finite volume methods, although we use them in a rather different context. The incomplete polynomial space replaces classical complete polynomial spaces in discrete formulations inspired by discontinuous Galerkin methods. Two problems are studied in this work: a heterogeneous anisotropic diffusion problem, which is used to lay the pillars of the method, and the incompressible Navier-Stokes equations, which provide a more realistic application. An exhaustive theoretical study as well as a set of numerical examples featuring different difficulties are provided.



Hide All
[1] Aavatsmark, I., Barkve, T., Bøe, Ø. and Mannseth, T., Discretization on unstructured grids for inhomogeneous, anisotropic media, Part I: Derivation of the methods. SIAM J. Sci. Comput. 19 (1998) 17001716.
[2] Aavatsmark, I., Barkve, T., Bøe, Ø. and Mannseth, T., Discretization on unstructured grids for inhomogeneous, anisotropic media, Part II: Discussion and numerical results. SIAM J. Sci. Comput. 19 (1998) 17171736.
[3] Aavatsmark, I., Eigestad, G.T., Mallison, B.T. and Nordbotten, J.M., A compact multipoint flux approximation method with improved robustness. Numer. Methods Partial Differential Equations 24 (2008) 13291360.
[4] Agélas, L., Di Pietro, D.A. and Droniou, J., The G method for heterogeneous anisotropic diffusion on general meshes. ESAIM: M2AN 44 (2010) 597625.
[5] Agélas, L., Di Pietro, D.A., Eymard, R. and Masson, R., An abstract analysis framework for nonconforming approximations of diffusion problems on general meshes. IJFV 7 (2010) 129.
[6] Arnold, D.N., An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742760.
[7] Arnold, D.N., Brezzi, F., Cockburn, B. and Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 17491779.
[8] J.-P. Aubin, Analyse fonctionnelle appliquée. Presses Universitaires de France, Paris (1987).
[9] Botti, L. and Di Pietro, D.A., A pressure-correction scheme for convection-dominated incompressible flows with discontinuous velocity and continuous pressure. J. Comput. Phys. 230 (2011) 572585.
[10] S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, 3th edition 15. Springer, New York (2008).
[11] Brezzi, F., Lipnikov, K. and Shashkov, M., Convergence of mimetic finite difference methods for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 45 (2005) 18721896.
[12] Brezzi, F., Lipnikov, K. and Simoncini, V., A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15 (2005) 15331553.
[13] Brezzi, F., Manzini, G., Marini, L.D., Pietra, P. and Russo, A., Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differential Equations 16 (2000) 365378.
[14] Buffa, A. and Ortner, C., Compact embeddings of broken Sobolev spaces and applications. IMA J. Numer. Anal. 4 (2009) 827855.
[15] Burman, E. and Ern, A., Continuous interior penalty $hp$-finite element methods for advection and advection-diffusion equations. Math. Comp. 76 (2007) 11191140.
[16] Burman, E. and Zunino, P., A domain decomposition method for partial differential equations with non-negative form based on interior penalties. SIAM J. Numer. Anal. 44 (2006) 16121638.
[17] Cao, Y., Helmig, R. and Wohlmuth, B.I., Geometrical interpretation of the multi-point flux approximation L-method. Internat. J. Numer. Methods Fluids 60 (2009) 11731199.
[18] P.G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)].
[19] Di Pietro, D.A., Analysis of a discontinuous Galerkin approximation of the Stokes problem based on an artificial compressibility flux. Internat. J. Numer. Methods Fluids 55 (2007) 793813.
[20] D.A. Di Pietro, Cell centered Galerkin methods. C. R. Acad. Sci. Paris, Sér. I 348 (2010) 31–34.
[21] D.A. Di Pietro, A compact cell-centered Galerkin method with subgrid stabilization. C. R. Acad. Sci. Paris, Sér. I 349 (2011) 93–98.
[22] Di Pietro, D.A. and Ern, A., Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations. Math. Comp. 79 (2010) 13031330.
[23] D.A. Di Pietro and A. Ern, Analysis of a discontinuous Galerkin method for heterogeneous diffusion problems with low-regularity solutions. Numer. Methods Partial Differential Equations (2011). Published online, DOI: 10.1002/num.20675.
[24] D.A. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods, Mathematics and Applications 69. Springer-Verlag, Berlin (2011). In press.
[25] Di Pietro, D.A., Ern, A. and Guermond, J.-L., Discontinuous Galerkin methods for anisotropic semi-definite diffusion with advection. SIAM J. Numer. Anal. 46 (2008) 805831.
[26] Droniou, J. and Eymard, R., A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math. 105 (2006) 3571.
[27] Droniou, J., Eymard, R., Gallouët, T. and Herbin, R., A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci. 20 (2010) 265295.
[28] M.G. Edwards and C.F. Rogers, A flux continuous scheme for the full tensor pressure equation, in Proc. of the 4th European Conf. on the Mathematics of Oil Recovery. D Røros, Norway (1994).
[29] Edwards, M.G. and Rogers, C.F., Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2 (1998) 259290.
[30] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences 159. Springer-Verlag, New York, NY (2004).
[31] Erturk, E., Corke, T.C. and Gökçöl, C., Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers. Internat. J. Numer. Methods Fluids 48 (2005) 747774.
[32] R. Eymard, Th. Gallouët and R. Herbin, The Finite Volume Method, edited by Ph. Charlet and J.L. Lions. North Holland (2000).
[33] Eymard, R., Gallouët, Th. and 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 (2010) 10091043.
[34] Eymard, R., Herbin, R. and Latché, J.-C., Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2D or 3D meshes. SIAM J. Numer. Anal. 45 (2007) 136.
[35] P. Grisvard, Singularities in Boundary Value Problems. Masson, Paris (1992).
[36] Heinrich, B. and Pietsch, K., Nitsche type mortaring for some elliptic problem with corner singularities. Computing 68 (2002) 217238.
[37] R. Herbin and F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, in Finite Volumes for Complex Applications V, edited by R. Eymard and J.-M. Hérard. John Wiley & Sons (2008) 659–692.
[38] R.B. Kellogg, On the Poisson equation with intersecting interfaces. Appl. Anal. 4 (1974/75) 101–129. Collection of articles dedicated to Nikolai Ivanovich Muskhelishvili.
[39] Kovasznay, L.S.G., Laminar flow behind a two-dimensional grid. Proc. Camb. Philos. Soc. 44 (1948) 5862.
[40] Nicaise, S. and Sändig, A.-M., General interface problems. I, II. Math. Methods Appl. Sci. 17 (1994) 395429, 431–450.
[41] J. Nitsche, On Dirichlet problems using subspaces with nearly zero boundary conditions, in The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972). Academic Press, New York (1972) 603–627.
[42] R. Stenberg, Mortaring by a method of J.A. Nitsche, in Computational Mechanics: New trends and applications, edited by S.R. Idelsohn, E. Oñate and E.N. Dvorkin. Barcelona, Spain (1998) 1–6. Centro Internacional de Métodos Numéricos en Ingeniería.
[43] R. Temam, Navier-Stokes Equations, Studies in Mathematics and its Applications 2. North-Holland Publishing Co., Amsterdam, revised edition (1979). Theory and numerical analysis, with an appendix by F. Thomasset.


Related content

Powered by UNSILO

Cell centered Galerkin methods for diffusive problems

  • Daniele A. Di Pietro (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.