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On the second-order convergence of a functionreconstructed from finite volume approximations of the Laplace equation onDelaunay-Voronoi meshes

Published online by Cambridge University Press:  30 November 2010

Pascal Omnes
Pascal Omnes*
Affiliation:
CEA, DEN, DM2S-SFME, 91191 Gif-sur-Yvette Cedex, France. [email protected] Université Paris 13, LAGA, CNRS UMR 7539, Institut Galilée, 99 avenue J.-B. Clément, 93430 Villetaneuse Cedex, France.
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Abstract

Cell-centered and vertex-centered finite volume schemes for the Laplace equationwith homogeneous Dirichlet boundary conditionsare considered on a triangular mesh and on the Voronoi diagram associated to its vertices.A broken P 1 function is constructed from the solutions of both schemes.When the domain is two-dimensional polygonal convex,it is shown that this reconstructionconverges with second-order accuracy towards the exact solution in the L 2 norm,under the sufficient condition that the right-hand side of the Laplace equation belongs to H 1(Ω).

Keywords

Finite volume methodLaplace equationDelaunay meshesVoronoi meshesconvergenceerror estimates
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 4 , July 2011 , pp. 627 - 650
Copyright
© EDP Sciences, SMAI, 2010

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References

Aavatsmark, I., Barkve, T., Bøe, Ø. and Mannseth, T., Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods. SIAM J. Sci. Comput. 19 (1998) 1700–1716. CrossRef
Aavatsmark, I., Barkve, T., Bøe, Ø. and Mannseth, T., Discretization on unstructured grids for inhomogeneous, anisotropic media. II. Discussion and numerical results. SIAM J. Sci. Comput. 19 (1998) 1717–1736. CrossRef
Andreianov, B., Boyer, F. and Hubert, F., Discrete duality finite volume schemes for Leray-Lions-type elliptic problems on general 2D meshes. Numer. Methods Partial Differ. Equ. 23 (2007) 145–195. CrossRef
Angermann, L., Numerical solution of second-order elliptic equations on plane domains. RAIRO Modél. Math. Anal. Numér. 25 (1991) 169–191. CrossRef
Bank, R.E. and Rose, D.J., Some error estimates for the box method. SIAM J. Numer. Anal. 24 (1987) 777–787. CrossRef
Bertolazzi, E. and Manzini, G., On vertex reconstructions for cell-centered finite volume approximations of 2D anisotropic diffusion problems. Math. Models Methods Appl. Sci. 17 (2007) 1–32. CrossRef
Boivin, S., Cayré, F. and Hérard, J.-M., Finite Volume, A method to solve the Navier Stokes equations for incompressible flows on unstructured meshes. Int. J. Thermal Sciences 39 (2000) 806–825. CrossRef
Boyer, F. and Hubert, F., Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities. SIAM J. Numer. Anal. 46 (2008) 3032–3070. CrossRef
Breil, J. and Maire, P.-H., A cell-centered diffusion scheme on two-dimensional unstructured meshes. J. Comput. Phys. 224 (2007) 785–823. CrossRef
Cai, Z., On the finite volume element method. Numer. Math. 58 (1991) 713–735. CrossRef
Cai, Z., Mandel, J. and McCormick, S., The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal. 28 (1991) 392–402. CrossRef
Carstensen, C., Lazarov, R. and Tomov, S., Explicit and averaging a posteriori error estimates for adaptive finite volume methods. SIAM J. Numer. Anal. 42 (2005) 2496–2521. CrossRef
Chainais-Hillairet, C., Discrete duality finite volume schemes for two-dimensional drift-diffusion and energy-transport models. Internat. J. Numer. Methods Fluids 59 (2009) 239–257. CrossRef
Chou, S.H., Kwak, D.Y. and Li, Q., L p error estimates and superconvergence for covolume or finite volume element methods. Numer. Methods Partial Differ. Equ. 19 (2003) 463–486. CrossRef
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
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).
Crouzeix, M. and Raviart, P.-A., 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.
Delcourte, S., Domelevo, K. and Omnes, P., 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. CrossRef
Domelevo, K. and Omnes, P., A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM: M2AN 39 (2005) 1203–1249. CrossRef
Ewing, R., Lazarov, R. and Lin, Y., Finite volume element approximations of nonlocal reactive flows in porous media. Numer. Methods Partial Differ. Equ. 16 (2000) 285–311. 3.0.CO;2-3>CrossRef
Ewing, R.E., Lin, T. and Lin, Y., On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numer. Anal. 39 (2002) 1865–1888. CrossRef
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.
Forsyth, P.A. and Sammon, P.H., Quadratic convergence for cell-centered grids. Appl. Numer. Math. 4 (1988) 377–394. CrossRef
Hackbucsh, W., On first and second order box schemes. Computing 41 (1989) 277–296. CrossRef
Herbin, R., 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. CrossRef
Hermeline, F., A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys. 160 (2000) 481–499. CrossRef
Hermeline, F., Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Comput. Methods Appl. Mech. Eng. 192 (2003) 1939–1959. CrossRef
Lazarov, R.D., Mishev, I.D. and Vassilevski, P.S., Finite volume methods for convection-diffusion problems. SIAM J. Numer. Anal. 33 (1996) 31–55. CrossRef
Le Potier, C., Finite volume scheme for highly anisotropic diffusion operators on unstructured meshes. C. R. Math. Acad. Sci. Paris 340 (2005) 921–926. CrossRef
Le Potier, C., Finite volume monotone scheme for highly anisotropic diffusion operators on unstructured triangular meshes. C. R. Math. Acad. Sci. Paris 341 (2005) 787–792. CrossRef
C. Le Potier, A nonlinear finite volume scheme satisfying maximum and minimum principles for diffusion operators. International Journal on Finite Volumes 6 (2009).
Mishev, I.D., Finite volume methods on Voronoi meshes. Numer. Methods Partial Differ. Equ. 14 (1998) 193–212. 3.0.CO;2-J>CrossRef
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).
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).
Süli, E., Convergence of finite volume schemes for Poisson's equation on nonuniform meshes. SIAM J. Numer. Anal. 28 (1991) 1419–1430. CrossRef
Vanselow, R. and Scheffler, H.P., Convergence analysis of a finite volume method via a new nonconforming finite element method. Numer. Methods Partial Differ. Equ. 14 (1998) 213–231. 3.0.CO;2-R>CrossRef
Weiser, A. and Wheeler, M.F., On convergence of block centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25 (1988) 351–375. CrossRef