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

Domelevo, Komla; Omnes, Pascal

ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 6, pp. 1203-1249.

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 $P^1$ 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 ${\mathrm{H}}_0^1$ norm and in the ${\mathrm{L}}^2$ 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 ${\mathrm{L}}^2$ 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.

[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. | Zbl

[2] I. Babuška and A.K. Aziz, On the angle condition in the finite element method. SIAM J. Numer. Anal. 13 (1976) 214-226. | Zbl

[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. | Numdam | Zbl

[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. | Zbl

[6] Y. Coudière, J.-P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. ESAIM: M2AN 33 (1999) 493-516. | Numdam | Zbl

[7] Y. Coudière and P. Villedieu, Convergence rate of a finite volume scheme for the linear convection-diffusion equation on locally refined meshes. ESAIM: M2AN 34 (2000) 1123-1149. | Numdam | Zbl

[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. | Zbl

[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. | Zbl

[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. | Zbl

[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. | Zbl

[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 Math 338 (2004) 741-746. | Zbl

[14] R. Herbin, 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. | Zbl

[15] F. Hermeline, A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys. 160 (2000) 481-499. | Zbl

[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. | Zbl

[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. | Zbl

[18] P. Jamet, Estimations d'erreur pour des éléments finis droits presque dégénérés. RAIRO Anal. numér. 10 (1976) 43-61. | Numdam | Zbl

[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. Fluids 33 (2004) 643-654. | Zbl

[20] I.D. Mishev, Finite volume methods on Voronoi meshes. Numer. Methods Partial Differential Equations 14 (1998) 193-212. | Zbl

[21] L.E. Payne and H.F. Weinberger, An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal. 5 (1960) 286-292. | Zbl

[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. | Zbl

[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. | Zbl

[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. | Zbl

[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 Equations 14 (1998) 213-231. | Zbl

[26] Special issue on the simulation of transport around a nuclear waste disposal site: the Couplex test cases. Computat. Geosci. 8 (2004). | Zbl