Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-28T04:45:20.308Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  15 November 2005

Komla Domelevo  and
Pascal Omnes
Komla Domelevo
Affiliation:
Mathématiques pour l'Industrie et la Physique, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France. komla@mip.ups-tlse.fr
Pascal Omnes
Affiliation:
Commissariat à l'Énergie Atomique, DEN-DM2S-SFME, 91191 Gif-sur-Yvette Cedex, France. pascal.omnes@cea.fr
Get access

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.

Keywords

Finite volume methodnon-conforming finite element methodLaplace equationdiscrete Green formuladiamond-cellerror estimatesconvergencesuperconvergencearbitrary meshesdegenerating meshesnon-conforming meshes.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 6 , November 2005 , pp. 1203 - 1249
Copyright
© EDP Sciences, SMAI, 2005

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

Acosta, G. and Durán, R.G., The maximum angle condition for mixed and nonconforming elements: application to the Stokes equations. SIAM J. Numer. Anal. 37 (1999) 18–36. CrossRef
Babuška, I. and Aziz, A.K., On the angle condition in the finite element method. SIAM J. Numer. Anal. 13 (1976) 214–226. CrossRef
Baranger, J., Maitre, J.-F. and Oudin, F., Connection between finite volume and mixed finite element methods. RAIRO Modél. Math. Anal Numér. 30 (1996) 445–465. CrossRef
Boivin, S., Cayré, F. and Hérard, J.-M., A finite volume method to solve the Navier-Stokes equations for incompressible flows on unstructured meshes. Int. J. Therm. Sci. 39 (2000) 806–825. CrossRef
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.
Coudière, Y., Vila, J.-P. and Villedieu, P., Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. ESAIM: M2AN 33 (1999) 493–516. CrossRef
Coudière, Y. and Villedieu, P., Convergence rate of a finite volume scheme for the linear convection-diffusion equation on locally refined meshes. ESAIM: M2AN 34 (2000) 1123–1149. CrossRef
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).
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.
Eymard, R., Gallouët, T. and Herbin, R., Finite volume approximation of elliptic problems and convergence of an approximate gradient. Appl. Numer. Math. 37 (2001) 31–53. CrossRef
Faille, I., A control volume method to solve an elliptic equation on a two-dimensional irregular meshing. Comput. Methods Appl. Mech. Engrg. 100 (1991) 275–290. CrossRef
Gallouët, T., Herbin, R. and Vignal, M.-H., Error estimates for the approximate finite volume solution of convection diffusion equations with general boundary conditions. SIAM J. Numer. Anal. 37 (2000) 1935–1972. CrossRef
Glowinski, R., He, J., Rappaz, J. and Wagner, J., A multi-domain method for solving numerically multi-scale elliptic problems. C. R. Acad. Sci. Paris Ser. I Math 338 (2004) 741–746. CrossRef
Herbin, R., An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh. Numer. Methods Partial Differential Equations 11 (1995) 165–173. CrossRef
Hermeline, F., A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys. 160 (2000) 481–499. CrossRef
Hyman, J.M. and Shashkov, M., Adjoint operators for the natural discretizations of the divergence, gradient, and curl on logically rectangular grids. Appl. Numer. Math. 25 (1997) 413–442. CrossRef
Hyman, J.M. and Shashkov, M., Natural discretizations for the divergence, gradient, and curl on logically rectangular grids. Comput. Math. Appl. 33 (1997) 81–104. CrossRef
Jamet, P., Estimations d'erreur pour des éléments finis droits presque dégénérés. RAIRO Anal. numér. 10 (1976) 43–61.
L. Klinger, J.B. Vos and K. Appert, A simplified gradient evaluation on non-orthogonal meshes; application to a plasma torch simulation method. Comput. Fluids 33 (2004) 643–654.
Mishev, I.D., Finite volume methods on Voronoi meshes. Numer. Methods Partial Differential Equations 14 (1998) 193–212. 3.0.CO;2-J>CrossRef
Payne, L.E. and Weinberger, H.F., An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal. 5 (1960) 286–292. CrossRef
Raviart, P.-A. and Thomas, J.-M., 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. CrossRef
Saas, L., Faille, I., Nataf, F. and Willien, F., Domain decomposition for a finite volume method on non-matching grids. C. R. Acad. Sci. Paris Ser. I Math. 338 (2004) 407–412. CrossRef
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.
Vanselow, R. and Scheffler, H.P., Convergence analysis of a finite volume method via a new nonconforming finite element method. Numer. Methods Partial Differential Equations 14 (1998) 213–231. 3.0.CO;2-R>CrossRef
Special issue on the simulation of transport around a nuclear waste disposal site: the Couplex test cases. Computat. Geosci. 8 (2004).