Cell centered Galerkin methods for diffusive problems

Daniele A. Di Pietro

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2012)

  • Volume: 46, Issue: 1, page 111-144
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Di Pietro, Daniele A.. "Cell centered Galerkin methods for diffusive problems." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 46.1 (2012): 111-144. <http://eudml.org/doc/273335>.

@article{DiPietro2012,
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.},
author = {Di Pietro, Daniele A.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {cell centered Galerkin; finite volumes; discontinuous Galerkin; heterogeneous anisotropic diffusion; incompressible Navier-Stokes equations; cell centered Galerkin method; discontinuous Galerkin method; numerical examples; finite element; convergence},
language = {eng},
number = {1},
pages = {111-144},
publisher = {EDP-Sciences},
title = {Cell centered Galerkin methods for diffusive problems},
url = {http://eudml.org/doc/273335},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Di Pietro, Daniele A.
TI - Cell centered Galerkin methods for diffusive problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2012
PB - EDP-Sciences
VL - 46
IS - 1
SP - 111
EP - 144
AB - 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.
LA - eng
KW - cell centered Galerkin; finite volumes; discontinuous Galerkin; heterogeneous anisotropic diffusion; incompressible Navier-Stokes equations; cell centered Galerkin method; discontinuous Galerkin method; numerical examples; finite element; convergence
UR - http://eudml.org/doc/273335
ER -

References

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  1. [1] I. Aavatsmark, T. Barkve, Ø. Bøe and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media, Part I: Derivation of the methods. SIAM J. Sci. Comput. 19 (1998) 1700–1716. Zbl0951.65080MR1618761
  2. [2] I. Aavatsmark, T. Barkve, Ø. Bøe and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media, Part II: Discussion and numerical results. SIAM J. Sci. Comput. 19 (1998) 1717–1736. Zbl0951.65082MR1611742
  3. [3] I. Aavatsmark, G.T. Eigestad, B.T. Mallison and J.M. Nordbotten, A compact multipoint flux approximation method with improved robustness. Numer. Methods Partial Differential Equations24 (2008) 1329–1360. Zbl1230.65114MR2427194
  4. [4] L. Agélas, D.A. Di Pietro and J. Droniou, The G method for heterogeneous anisotropic diffusion on general meshes. ESAIM: M2AN 44 (2010) 597–625. Zbl1202.65143MR2683575
  5. [5] L. Agélas, D.A. Di Pietro, R. Eymard and R. Masson, An abstract analysis framework for nonconforming approximations of diffusion problems on general meshes. IJFV7 (2010) 1–29. MR2595777
  6. [6] D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal.19 (1982) 742–760. Zbl0482.65060MR664882
  7. [7] D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal.39 (2002) 1749–1779. Zbl1008.65080MR1885715
  8. [8] J.-P. Aubin, Analyse fonctionnelle appliquée. Presses Universitaires de France, Paris (1987). Zbl0654.46001
  9. [9] L. Botti and D.A. Di Pietro, A pressure-correction scheme for convection-dominated incompressible flows with discontinuous velocity and continuous pressure. J. Comput. Phys.230 (2011) 572–585. Zbl1283.76030MR2745444
  10. [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). Zbl1135.65042MR2373954
  11. [11] F. Brezzi, K. Lipnikov and M. Shashkov, Convergence of mimetic finite difference methods for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal.45 (2005) 1872–1896. Zbl1108.65102MR2192322
  12. [12] F. Brezzi, K. Lipnikov and V. Simoncini, A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci.15 (2005) 1533–1553. Zbl1083.65099MR2168945
  13. [13] F. Brezzi, G. Manzini, L.D. Marini, P. Pietra and A. Russo, Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differential Equations16 (2000) 365–378. Zbl0957.65099MR1765651
  14. [14] A. Buffa and C. Ortner, Compact embeddings of broken Sobolev spaces and applications. IMA J. Numer. Anal.4 (2009) 827–855. Zbl1181.65094MR2557047
  15. [15] E. Burman and A. Ern, Continuous interior penalty h p -finite element methods for advection and advection-diffusion equations. Math. Comp.76 (2007) 1119–1140. Zbl1118.65118MR2299768
  16. [16] E. Burman and P. Zunino, A domain decomposition method for partial differential equations with non-negative form based on interior penalties. SIAM J. Numer. Anal.44 (2006) 1612–1638. Zbl1125.65113MR2257119
  17. [17] Y. Cao, R. Helmig and B.I. Wohlmuth, Geometrical interpretation of the multi-point flux approximation L-method. Internat. J. Numer. Methods Fluids60 (2009) 1173–1199. Zbl1166.76042MR2554072
  18. [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)]. Zbl0999.65129MR1930132
  19. [19] D.A. Di Pietro, Analysis of a discontinuous Galerkin approximation of the Stokes problem based on an artificial compressibility flux. Internat. J. Numer. Methods Fluids55 (2007) 793–813. Zbl1128.76034MR2359551
  20. [20] D.A. Di Pietro, Cell centered Galerkin methods. C. R. Acad. Sci. Paris, Sér. I 348 (2010) 31–34. Zbl1183.65123MR2586739
  21. [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. Zbl1208.65165MR2755705
  22. [22] D.A. Di Pietro and A. Ern, Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations. Math. Comp.79 (2010) 1303–1330. Zbl05776268MR2629994
  23. [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. Zbl1267.65178MR2914787
  24. [24] D.A. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods, Mathematics and Applications 69. Springer-Verlag, Berlin (2011). In press. Zbl1231.65209
  25. [25] D.A. Di Pietro, A. Ern and J.-L. Guermond, Discontinuous Galerkin methods for anisotropic semi-definite diffusion with advection. SIAM J. Numer. Anal.46 (2008) 805–831. Zbl1165.49032MR2383212
  26. [26] J. Droniou and R. Eymard, A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math.105 (2006) 35–71. Zbl1109.65099MR2257385
  27. [27] J. Droniou, R. Eymard, T. Gallouët and R. Herbin, A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci.20 (2010) 265–295. Zbl1191.65142MR2649153
  28. [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. [29] M.G. Edwards and C.F. Rogers, Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci.2 (1998) 259–290. Zbl0945.76049MR1686429
  30. [30] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences 159. Springer-Verlag, New York, NY (2004). Zbl1059.65103MR2050138
  31. [31] E. Erturk, T.C. Corke and C. Gökçöl, Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers. Internat. J. Numer. Methods Fluids48 (2005) 747–774. Zbl1071.76038
  32. [32] R. Eymard, Th. Gallouët and R. Herbin, The Finite Volume Method, edited by Ph. Charlet and J.L. Lions. North Holland (2000). Zbl0981.65095MR1804748
  33. [33] R. Eymard, Th. Gallouët and R. Herbin, 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. Zbl1202.65144MR2727814
  34. [34] R. Eymard, R. Herbin and J.-C. Latché, 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. Zbl1173.76028MR2285842
  35. [35] P. Grisvard, Singularities in Boundary Value Problems. Masson, Paris (1992). Zbl0766.35001MR1173209
  36. [36] B. Heinrich and K. Pietsch, Nitsche type mortaring for some elliptic problem with corner singularities. Computing68 (2002) 217–238. Zbl1002.65124MR1914113
  37. [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. Zbl1246.76053MR2451465
  38. [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. Zbl0307.35038MR393815
  39. [39] L.S.G. Kovasznay, Laminar flow behind a two-dimensional grid. Proc. Camb. Philos. Soc.44 (1948) 58–62. Zbl0030.22902
  40. [40] S. Nicaise and A.-M. Sändig, General interface problems. I, II. Math. Methods Appl. Sci. 17 (1994) 395–429, 431–450. Zbl0824.35015MR1274152
  41. [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. Zbl0271.65059MR426456
  42. [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. Zbl0970.74003MR1839048
  43. [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. Zbl0426.35003MR603444

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