On the second-order convergence of a function reconstructed from finite volume approximations of the Laplace equation on Delaunay-Voronoi meshes

Omnes, Pascal

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 4, pp. 627-650.

Cell-centered and vertex-centered finite volume schemes for the Laplace equation with homogeneous Dirichlet boundary conditions are considered on a triangular mesh and on the Voronoi diagram associated to its vertices. A broken P¹ function is constructed from the solutions of both schemes. When the domain is two-dimensional polygonal convex, it is shown that this reconstruction converges with second-order accuracy towards the exact solution in the L² norm, under the sufficient condition that the right-hand side of the Laplace equation belongs to H¹(Ω).

Classification : 65N15, 65N30, 35J05
Mots clés : finite volume method, Laplace equation, Delaunay meshes, Voronoi meshes, convergence, error estimates

[1] I. Aavatsmark, T. Barkve, Ø. Bøe and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods. SIAM J. Sci. Comput. 19 (1998) 1700-1716. | MR | Zbl

[2] I. Aavatsmark, T. Barkve, Ø. Bøe and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media. II. Discussion and numerical results. SIAM J. Sci. Comput. 19 (1998) 1717-1736. | MR | Zbl

[3] B. Andreianov, F. Boyer and F. Hubert, Discrete duality finite volume schemes for Leray-Lions-type elliptic problems on general 2D meshes. Numer. Methods Partial Differ. Equ. 23 (2007) 145-195. | MR | Zbl

[4] L. Angermann, Numerical solution of second-order elliptic equations on plane domains. RAIRO Modél. Math. Anal. Numér. 25 (1991) 169-191. | Numdam | MR | Zbl

[5] R.E. Bank and D.J. Rose, Some error estimates for the box method. SIAM J. Numer. Anal. 24 (1987) 777-787. | MR | Zbl

[6] E. Bertolazzi and G. Manzini, On vertex reconstructions for cell-centered finite volume approximations of 2D anisotropic diffusion problems. Math. Models Methods Appl. Sci. 17 (2007) 1-32. | MR | Zbl

[7] 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. Thermal Sciences 39 (2000) 806-825.

[8] F. Boyer and F. Hubert, Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities. SIAM J. Numer. Anal. 46 (2008) 3032-3070. | MR | Zbl

[9] J. Breil and P.-H. Maire, A cell-centered diffusion scheme on two-dimensional unstructured meshes. J. Comput. Phys. 224 (2007) 785-823. | MR | Zbl

[10] Z. Cai, On the finite volume element method. Numer. Math. 58 (1991) 713-735. | MR | Zbl

[11] Z. Cai, J. Mandel and S. Mccormick, The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal. 28 (1991) 392-402. | MR | Zbl

[12] C. Carstensen, R. Lazarov and S. Tomov, Explicit and averaging a posteriori error estimates for adaptive finite volume methods. SIAM J. Numer. Anal. 42 (2005) 2496-2521. | MR | Zbl

[13] C. Chainais-Hillairet, Discrete duality finite volume schemes for two-dimensional drift-diffusion and energy-transport models. Internat. J. Numer. Methods Fluids 59 (2009) 239-257. | MR | Zbl

[14] S.H. Chou, D.Y. Kwak and Q. Li, Lp error estimates and superconvergence for covolume or finite volume element methods. Numer. Methods Partial Differ. Equ. 19 (2003) 463-486. | MR | Zbl

[15] 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 | MR | Zbl

[16] Y. Coudière, C. Pierre, O. Rousseau and R. Turpault, A 2D/3D Discrete Duality Finite Volume Scheme. Application to ECG simulation. International Journal on Finite Volumes 6 (2009). | MR

[17] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973) 33-75. | Numdam | MR | Zbl

[18] S. Delcourte, K. Domelevo and P. Omnes, A discrete duality finite volume approach to Hodge decomposition and div-curl problems on almost arbitrary two-dimensional meshes. SIAM J. Numer. Anal. 45 (2007) 1142-1174. | MR | Zbl

[19] K. Domelevo and P. Omnes, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM: M2AN 39 (2005) 1203-1249. | Numdam | MR | Zbl

[20] R. Ewing, R. Lazarov and Y. Lin, Finite volume element approximations of nonlocal reactive flows in porous media. Numer. Methods Partial Differ. Equ. 16 (2000) 285-311. | MR | Zbl

[21] R.E. Ewing, T. Lin and Y. Lin, On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numer. Anal. 39 (2002) 1865-1888. | MR | Zbl

[22] R. Eymard, T. Gallouët and R. Herbin, Handbook of numerical analysis 7, P.G. Ciarlet and J.-L. Lions Eds., North-Holland/Elsevier, Amsterdam (2000) 713-1020. | MR | Zbl

[23] P.A. Forsyth and P.H. Sammon, Quadratic convergence for cell-centered grids. Appl. Numer. Math. 4 (1988) 377-394. | MR | Zbl

[24] W. Hackbucsh, On first and second order box schemes. Computing 41 (1989) 277-296. | Zbl

[25] R. Herbin, An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh. Numer. Methods Partial Differ. Equ. 11 (1995) 165-173. | MR | Zbl

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

[27] F. Hermeline, Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Comput. Methods Appl. Mech. Eng. 192 (2003) 1939-1959. | MR | Zbl

[28] R.D. Lazarov, I.D. Mishev and P.S. Vassilevski, Finite volume methods for convection-diffusion problems. SIAM J. Numer. Anal. 33 (1996) 31-55. | MR | Zbl

[29] C. Le Potier, Finite volume scheme for highly anisotropic diffusion operators on unstructured meshes. C. R. Math. Acad. Sci. Paris 340 (2005) 921-926. | MR | Zbl

[30] C. Le Potier, Finite volume monotone scheme for highly anisotropic diffusion operators on unstructured triangular meshes. C. R. Math. Acad. Sci. Paris 341 (2005) 787-792. | MR | Zbl

[31] C. Le Potier, A nonlinear finite volume scheme satisfying maximum and minimum principles for diffusion operators. International Journal on Finite Volumes 6 (2009). | MR

[32] I.D. Mishev, Finite volume methods on Voronoi meshes. Numer. Methods Partial Differ. Equ. 14 (1998) 193-212. | MR | Zbl

[33] A. Njifenjou and A.J. Kinfack, Convergence analysis of an MPFA method for flow problems in anisotropic heterogeneous porous media. International Journal on Finite Volumes 5 (2008). | MR

[34] P. Omnes, Error estimates for a finite volume method for the Laplace equation in dimension one through discrete Green functions. International Journal on Finite Volumes 6 (2009). | MR

[35] E. Süli, Convergence of finite volume schemes for Poisson's equation on nonuniform meshes. SIAM J. Numer. Anal. 28 (1991) 1419-1430. | MR | Zbl

[36] R. Vanselow and H.P. Scheffler, Convergence analysis of a finite volume method via a new nonconforming finite element method. Numer. Methods Partial Differ. Equ. 14 (1998) 213-231. | MR | Zbl

[37] A. Weiser and M.F. Wheeler, On convergence of block centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25 (1988) 351-375. | MR | Zbl