A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids

Komla Domelevo; Pascal Omnes

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 39, Issue: 6, page 1203-1249
  • ISSN: 0764-583X

Abstract

top
We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires “Voronoi-type” meshes. We show the equivalence of this finite volume method with a non-conforming finite element method with basis functions being P1 on the cells, generally called “diamond-cells”, of a third mesh. Under geometrical conditions on these diamond-cells, we prove a first-order convergence both in the 0 norm and in the L² norm. Superconvergence results are obtained on certain types of homothetically refined grids. Finally, numerical experiments confirm these results and also show second-order convergence in the L² norm on general grids. They also indicate that this method performs particularly well for the approximation of the gradient of the solution, and may be used on degenerating triangular grids. An example of application on non-conforming locally refined grids is given.

How to cite

top

Domelevo, Komla, and Omnes, Pascal. "A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids." ESAIM: Mathematical Modelling and Numerical Analysis 39.6 (2010): 1203-1249. <http://eudml.org/doc/194302>.

@article{Domelevo2010,
abstract = { We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires “Voronoi-type” meshes. We show the equivalence of this finite volume method with a non-conforming finite element method with basis functions being P1 on the cells, generally called “diamond-cells”, of a third mesh. Under geometrical conditions on these diamond-cells, we prove a first-order convergence both in the $\xHone_0$ norm and in the L² norm. Superconvergence results are obtained on certain types of homothetically refined grids. Finally, numerical experiments confirm these results and also show second-order convergence in the L² norm on general grids. They also indicate that this method performs particularly well for the approximation of the gradient of the solution, and may be used on degenerating triangular grids. An example of application on non-conforming locally refined grids is given. },
author = {Domelevo, Komla, Omnes, Pascal},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite volume method; non-conforming finite element method; Laplace equation; discrete Green formula; diamond-cell; error estimates; convergence; superconvergence; arbitrary meshes; degenerating meshes; non-conforming meshes.; nonconforming finite element method; error estimate; nonconforming meshes; finite volume method; Laplace equation; numerical experiments},
language = {eng},
month = {3},
number = {6},
pages = {1203-1249},
publisher = {EDP Sciences},
title = {A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids},
url = {http://eudml.org/doc/194302},
volume = {39},
year = {2010},
}

TY - JOUR
AU - Domelevo, Komla
AU - Omnes, Pascal
TI - A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 39
IS - 6
SP - 1203
EP - 1249
AB - We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires “Voronoi-type” meshes. We show the equivalence of this finite volume method with a non-conforming finite element method with basis functions being P1 on the cells, generally called “diamond-cells”, of a third mesh. Under geometrical conditions on these diamond-cells, we prove a first-order convergence both in the $\xHone_0$ norm and in the L² norm. Superconvergence results are obtained on certain types of homothetically refined grids. Finally, numerical experiments confirm these results and also show second-order convergence in the L² norm on general grids. They also indicate that this method performs particularly well for the approximation of the gradient of the solution, and may be used on degenerating triangular grids. An example of application on non-conforming locally refined grids is given.
LA - eng
KW - Finite volume method; non-conforming finite element method; Laplace equation; discrete Green formula; diamond-cell; error estimates; convergence; superconvergence; arbitrary meshes; degenerating meshes; non-conforming meshes.; nonconforming finite element method; error estimate; nonconforming meshes; finite volume method; Laplace equation; numerical experiments
UR - http://eudml.org/doc/194302
ER -

References

top
  1. G. Acosta and R.G. Durán, The maximum angle condition for mixed and nonconforming elements: application to the Stokes equations. SIAM J. Numer. Anal.37 (1999) 18–36.  
  2. I. Babuška and A.K. Aziz, On the angle condition in the finite element method. SIAM J. Numer. Anal.13 (1976) 214–226.  
  3. J. Baranger, J.-F. Maitre and F. Oudin, Connection between finite volume and mixed finite element methods. RAIRO Modél. Math. Anal Numér.30 (1996) 445–465.  
  4. S. Boivin, F. Cayré and J.-M. Hérard, A finite volume method to solve the Navier-Stokes equations for incompressible flows on unstructured meshes. Int. J. Therm. Sci.39 (2000) 806–825.  
  5. P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis Vol. 2, P.G. Ciarlet and J.-L. Lions, Eds., Amsterdam North-Holland/Elsevier (1991) 17–351.  
  6. Y. Coudière, J.-P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. ESAIM: M2AN33 (1999) 493–516.  
  7. Y. Coudière and P. Villedieu, Convergence rate of a finite volume scheme for the linear convection-diffusion equation on locally refined meshes. ESAIM: M2AN34 (2000) 1123–1149.  
  8. K. Domelevo and P. Omnes, Construction et analyse numérique d'une méthode de volumes finis pour l'équation de Laplace sur des maillages bidimensionnels presque quelconques (in French), Rapport CEA (2004).  
  9. R. Eymard, T. Gallouët and R. Herbin, Handbook of Numerical Analysis Vol. 7, P.G. Ciarlet and J.-L. Lions, Eds., North-Holland/Elsevier, Amsterdam (2000) 713–1020.  
  10. R. Eymard, T. Gallouët and R. Herbin, Finite volume approximation of elliptic problems and convergence of an approximate gradient. Appl. Numer. Math.37 (2001) 31–53.  
  11. I. Faille, A control volume method to solve an elliptic equation on a two-dimensional irregular meshing. Comput. Methods Appl. Mech. Engrg.100 (1991) 275–290.  
  12. T. Gallouët, R. Herbin and M.-H. Vignal, Error estimates for the approximate finite volume solution of convection diffusion equations with general boundary conditions. SIAM J. Numer. Anal.37 (2000) 1935–1972.  
  13. R. Glowinski, J. He, J. Rappaz and J. Wagner, A multi-domain method for solving numerically multi-scale elliptic problems. C. R. Acad. Sci. Paris Ser. I Math338 (2004) 741–746.  
  14. R. Herbin, An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh. Numer. Methods Partial Differential Equations11 (1995) 165–173.  
  15. F. Hermeline, A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys.160 (2000) 481–499.  
  16. J.M. Hyman and M. Shashkov, Adjoint operators for the natural discretizations of the divergence, gradient, and curl on logically rectangular grids. Appl. Numer. Math.25 (1997) 413–442.  
  17. J.M. Hyman and M. Shashkov, Natural discretizations for the divergence, gradient, and curl on logically rectangular grids. Comput. Math. Appl.33 (1997) 81–104.  
  18. P. Jamet, Estimations d'erreur pour des éléments finis droits presque dégénérés. RAIRO Anal. numér.10 (1976) 43–61.  
  19. L. Klinger, J.B. Vos and K. Appert, A simplified gradient evaluation on non-orthogonal meshes; application to a plasma torch simulation method. Comput. Fluids33 (2004) 643–654.  
  20. I.D. Mishev, Finite volume methods on Voronoi meshes. Numer. Methods Partial Differential Equations14 (1998) 193–212.  
  21. L.E. Payne and H.F. Weinberger, An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal.5 (1960) 286–292.  
  22. P.-A. Raviart and J.-M. Thomas, A mixed finite element method for second order elliptic problems, in Mathematical aspects of the finite element method, I. Galligani and E. Magenes, Eds., Springer-Verlag, New-York. Lecture Notes in Math.606 (1977) 292–315.  
  23. L. Saas, I. Faille, F. Nataf and F. Willien, Domain decomposition for a finite volume method on non-matching grids. C. R. Acad. Sci. Paris Ser. I Math.338 (2004) 407–412.  
  24. G. Strang, Variational crimes in the finite element method, in The mathematical foundations of the finite element method with applications to partial differential equations, A.K. Aziz Ed., Academic Press, New York (1972) 689–710.  
  25. R. Vanselow and H.P. Scheffler, Convergence analysis of a finite volume method via a new nonconforming finite element method. Numer. Methods Partial Differential Equations14 (1998) 213–231.  
  26. Special issue on the simulation of transport around a nuclear waste disposal site: the Couplex test cases. Computat. Geosci.8 (2004).  

Citations in EuDML Documents

top
  1. Christophe Berthon, Yves Coudière, Vivien Desveaux, Second-order MUSCL schemes based on Dual Mesh Gradient Reconstruction (DMGR)
  2. Philippe Angot, Franck Boyer, Florence Hubert, Asymptotic and numerical modelling of flows in fractured porous media
  3. Robert Eymard, Cindy Guichard, Raphaèle Herbin, Small-stencil 3D schemes for diffusive flows in porous media
  4. Robert Eymard, Cindy Guichard, Raphaèle Herbin, Small-stencil 3D schemes for diffusive flows in porous media
  5. Jérôme Bonelle, Alexandre Ern, Analysis of Compatible Discrete Operator schemes for elliptic problems on polyhedral meshes
  6. Pascal Omnes, On the second-order convergence of a function reconstructed from finite volume approximations of the Laplace equation on Delaunay-Voronoi meshes
  7. Pascal Omnes, On the second-order convergence of a function reconstructed from finite volume approximations of the Laplace equation on Delaunay-Voronoi meshes

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.