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).
Keywords: finite volume method, Laplace equation, Delaunay meshes, Voronoi meshes, convergence, error estimates
@article{M2AN_2011__45_4_627_0,
author = {Omnes, Pascal},
title = {On the second-order convergence of a function reconstructed from finite volume approximations of the {Laplace} equation on {Delaunay-Voronoi} meshes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {627--650},
year = {2011},
publisher = {EDP Sciences},
volume = {45},
number = {4},
doi = {10.1051/m2an/2010068},
mrnumber = {2804653},
zbl = {1269.65109},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2010068/}
}
TY - JOUR AU - Omnes, Pascal TI - On the second-order convergence of a function reconstructed from finite volume approximations of the Laplace equation on Delaunay-Voronoi meshes JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2011 SP - 627 EP - 650 VL - 45 IS - 4 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2010068/ DO - 10.1051/m2an/2010068 LA - en ID - M2AN_2011__45_4_627_0 ER -
%0 Journal Article %A Omnes, Pascal %T On the second-order convergence of a function reconstructed from finite volume approximations of the Laplace equation on Delaunay-Voronoi meshes %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2011 %P 627-650 %V 45 %N 4 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2010068/ %R 10.1051/m2an/2010068 %G en %F M2AN_2011__45_4_627_0
Omnes, Pascal. On the second-order convergence of a function reconstructed from finite volume approximations of the Laplace equation on Delaunay-Voronoi meshes. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 4, pp. 627-650. doi: 10.1051/m2an/2010068
[1] , , and , Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods. SIAM J. Sci. Comput. 19 (1998) 1700-1716. | Zbl | MR
[2] , , and , Discretization on unstructured grids for inhomogeneous, anisotropic media. II. Discussion and numerical results. SIAM J. Sci. Comput. 19 (1998) 1717-1736. | Zbl | MR
[3] , and , Discrete duality finite volume schemes for Leray-Lions-type elliptic problems on general 2D meshes. Numer. Methods Partial Differ. Equ. 23 (2007) 145-195. | Zbl | MR
[4] , Numerical solution of second-order elliptic equations on plane domains. RAIRO Modél. Math. Anal. Numér. 25 (1991) 169-191. | Zbl | MR | Numdam
[5] and , Some error estimates for the box method. SIAM J. Numer. Anal. 24 (1987) 777-787. | Zbl | MR
[6] and , On vertex reconstructions for cell-centered finite volume approximations of 2D anisotropic diffusion problems. Math. Models Methods Appl. Sci. 17 (2007) 1-32. | Zbl | MR
[7] , and , 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] and , Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities. SIAM J. Numer. Anal. 46 (2008) 3032-3070. | Zbl | MR
[9] and , A cell-centered diffusion scheme on two-dimensional unstructured meshes. J. Comput. Phys. 224 (2007) 785-823. | Zbl | MR
[10] , On the finite volume element method. Numer. Math. 58 (1991) 713-735. | Zbl | MR
[11] , and , The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal. 28 (1991) 392-402. | Zbl | MR
[12] , and , Explicit and averaging a posteriori error estimates for adaptive finite volume methods. SIAM J. Numer. Anal. 42 (2005) 2496-2521. | Zbl | MR
[13] , Discrete duality finite volume schemes for two-dimensional drift-diffusion and energy-transport models. Internat. J. Numer. Methods Fluids 59 (2009) 239-257. | Zbl | MR
[14] , and , Lp error estimates and superconvergence for covolume or finite volume element methods. Numer. Methods Partial Differ. Equ. 19 (2003) 463-486. | Zbl | MR
[15] , and , Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. ESAIM: M2AN 33 (1999) 493-516. | Zbl | MR | Numdam
[16] , , and , A 2D/3D Discrete Duality Finite Volume Scheme. Application to ECG simulation. International Journal on Finite Volumes 6 (2009). | MR
[17] and , 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. | Zbl | MR | Numdam
[18] , and , 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. | Zbl | MR
[19] and , A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM: M2AN 39 (2005) 1203-1249. | Zbl | MR | Numdam
[20] , and , Finite volume element approximations of nonlocal reactive flows in porous media. Numer. Methods Partial Differ. Equ. 16 (2000) 285-311. | Zbl | MR
[21] , and , On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numer. Anal. 39 (2002) 1865-1888. | Zbl | MR
[22] , and , Handbook of numerical analysis 7, P.G. Ciarlet and J.-L. Lions Eds., North-Holland/Elsevier, Amsterdam (2000) 713-1020. | Zbl | MR
[23] and , Quadratic convergence for cell-centered grids. Appl. Numer. Math. 4 (1988) 377-394. | Zbl | MR
[24] , On first and second order box schemes. Computing 41 (1989) 277-296. | Zbl
[25] , 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. | Zbl | MR
[26] , A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys. 160 (2000) 481-499. | Zbl | MR
[27] , Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Comput. Methods Appl. Mech. Eng. 192 (2003) 1939-1959. | Zbl | MR
[28] , and , Finite volume methods for convection-diffusion problems. SIAM J. Numer. Anal. 33 (1996) 31-55. | Zbl | MR
[29] , Finite volume scheme for highly anisotropic diffusion operators on unstructured meshes. C. R. Math. Acad. Sci. Paris 340 (2005) 921-926. | Zbl | MR
[30] , Finite volume monotone scheme for highly anisotropic diffusion operators on unstructured triangular meshes. C. R. Math. Acad. Sci. Paris 341 (2005) 787-792. | Zbl | MR
[31] , A nonlinear finite volume scheme satisfying maximum and minimum principles for diffusion operators. International Journal on Finite Volumes 6 (2009). | MR
[32] , Finite volume methods on Voronoi meshes. Numer. Methods Partial Differ. Equ. 14 (1998) 193-212. | Zbl | MR
[33] and , Convergence analysis of an MPFA method for flow problems in anisotropic heterogeneous porous media. International Journal on Finite Volumes 5 (2008). | MR
[34] , 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] , Convergence of finite volume schemes for Poisson's equation on nonuniform meshes. SIAM J. Numer. Anal. 28 (1991) 1419-1430. | Zbl | MR
[36] and , Convergence analysis of a finite volume method via a new nonconforming finite element method. Numer. Methods Partial Differ. Equ. 14 (1998) 213-231. | Zbl | MR
[37] and , On convergence of block centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25 (1988) 351-375. | Zbl | MR
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