Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T05:25:00.759Z Has data issue: false hasContentIssue false
  • English
  • Français

Cell centered Galerkin methods for diffusive problems

Published online by Cambridge University Press:  24 August 2011

Daniele A. Di Pietro
Daniele A. Di Pietro*
Affiliation:
IFP Energies nouvelles, 1 & 4 avenue de Bois Préau, 92582 Rueil-Malmaison Cedex, France. daniale-antonio.di-pietro@ifpen.fr
Get access

Abstract

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.

Keywords

Cell centered Galerkinfinite volumesdiscontinuous Galerkinheterogeneous anisotropic diffusionincompressible Navier-Stokes equations
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 1 , January 2012 , pp. 111 - 144
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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) 1700–1716. CrossRef
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) 1717–1736. CrossRef
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) 1329–1360. CrossRef
Agélas, L., Di Pietro, D.A. and Droniou, J., The G method for heterogeneous anisotropic diffusion on general meshes. ESAIM: M2AN 44 (2010) 597–625. CrossRef
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) 1–29.
Arnold, D.N., An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742–760. CrossRef
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) 1749–1779. CrossRef
J.-P. Aubin, Analyse fonctionnelle appliquée. Presses Universitaires de France, Paris (1987).
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) 572–585. CrossRef
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).
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) 1872–1896. CrossRef
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) 1533–1553. CrossRef
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) 365–378. 3.0.CO;2-Y>CrossRef
Buffa, A. and Ortner, C., Compact embeddings of broken Sobolev spaces and applications. IMA J. Numer. Anal. 4 (2009) 827–855. CrossRef
Burman, E. and Ern, A., Continuous interior penalty $hp$-finite element methods for advection and advection-diffusion equations. Math. Comp. 76 (2007) 1119–1140. CrossRef
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) 1612–1638. CrossRef
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) 1173–1199. CrossRef
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)].
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) 793–813. CrossRef
D.A. Di Pietro, Cell centered Galerkin methods. C. R. Acad. Sci. Paris, Sér. I 348 (2010) 31–34.
D.A. Di Pietro, A compact cell-centered Galerkin method with subgrid stabilization. C. R. Acad. Sci. Paris, Sér. I 349 (2011) 93–98.
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) 1303–1330. CrossRef
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.
D.A. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods, Mathematics and Applications 69. Springer-Verlag, Berlin (2011). In press.
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) 805–831. CrossRef
Droniou, J. and Eymard, R., A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math. 105 (2006) 35–71. CrossRef
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) 265–295. CrossRef
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).
Edwards, M.G. and Rogers, C.F., Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2 (1998) 259–290. CrossRef
A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences 159. Springer-Verlag, New York, NY (2004).
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) 747–774. CrossRef
R. Eymard, Th. Gallouët and R. Herbin, The Finite Volume Method, edited by Ph. Charlet and J.L. Lions. North Holland (2000).
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) 1009–1043. CrossRef
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) 1–36. CrossRef
P. Grisvard, Singularities in Boundary Value Problems. Masson, Paris (1992).
Heinrich, B. and Pietsch, K., Nitsche type mortaring for some elliptic problem with corner singularities. Computing 68 (2002) 217–238. CrossRef
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.
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.
Kovasznay, L.S.G., Laminar flow behind a two-dimensional grid. Proc. Camb. Philos. Soc. 44 (1948) 58–62. CrossRef
Nicaise, S. and Sändig, A.-M., General interface problems. I, II. Math. Methods Appl. Sci. 17 (1994) 395–429, 431–450. CrossRef
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.
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.
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.