Cell centered Galerkin methods for diffusive problems

Daniele A. Di Pietro

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

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

Abstract

top
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

top

Di Pietro, Daniele A.. "Cell centered Galerkin methods for diffusive problems." ESAIM: Mathematical Modelling and Numerical Analysis 46.1 (2011): 111-144. <http://eudml.org/doc/222101>.

@article{DiPietro2011,
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},
keywords = {Cell centered Galerkin; finite volumes; discontinuous Galerkin; heterogeneous anisotropic diffusion; incompressible Navier-Stokes equations; cell centered Galerkin method; discontinuous Galerkin method; incompressible Navier-Stokes equations; numerical examples; finite element; convergence},
language = {eng},
month = {8},
number = {1},
pages = {111-144},
publisher = {EDP Sciences},
title = {Cell centered Galerkin methods for diffusive problems},
url = {http://eudml.org/doc/222101},
volume = {46},
year = {2011},
}

TY - JOUR
AU - Di Pietro, Daniele A.
TI - Cell centered Galerkin methods for diffusive problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/8//
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; incompressible Navier-Stokes equations; numerical examples; finite element; convergence
UR - http://eudml.org/doc/222101
ER -

References

top
  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.  
  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.  
  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.  
  4. L. Agélas, D.A. Di Pietro and J. Droniou, The G method for heterogeneous anisotropic diffusion on general meshes. ESAIM: M2AN44 (2010) 597–625.  
  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.  
  6. D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal.19 (1982) 742–760.  
  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.  
  8. J.-P. Aubin, Analyse fonctionnelle appliquée. Presses Universitaires de France, Paris (1987).  
  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.  
  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. 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.  
  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.  
  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.  
  14. A. Buffa and C. Ortner, Compact embeddings of broken Sobolev spaces and applications. IMA J. Numer. Anal.4 (2009) 827–855.  
  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.  
  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.  
  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.  
  18. P.G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)].  
  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.  
  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. 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.  
  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. 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.  
  26. J. Droniou and R. Eymard, A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math.105 (2006) 35–71.  
  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.  
  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. 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.  
  30. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences159. Springer-Verlag, New York, NY (2004).  
  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.  
  32. R. Eymard, Th. Gallouët and R. Herbin, The Finite Volume Method, edited by Ph. Charlet and J.L. Lions. North Holland (2000).  
  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.  
  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.  
  35. P. Grisvard, Singularities in Boundary Value Problems. Masson, Paris (1992).  
  36. B. Heinrich and K. Pietsch, Nitsche type mortaring for some elliptic problem with corner singularities. Computing68 (2002) 217–238.  
  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. L.S.G. Kovasznay, Laminar flow behind a two-dimensional grid. Proc. Camb. Philos. Soc.44 (1948) 58–62.  
  40. S. Nicaise and A.-M. Sändig, General interface problems. I, II. Math. Methods Appl. Sci.17 (1994) 395–429, 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 Applications2. North-Holland Publishing Co., Amsterdam, revised edition (1979). Theory and numerical analysis, with an appendix by F. Thomasset.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.