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

- Volume: 45, Issue: 4, page 627-650
- ISSN: 0764-583X

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topOmnes, 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 - Modélisation Mathématique et Analyse Numérique 45.4 (2011): 627-650. <http://eudml.org/doc/273094>.

@article{Omnes2011,

abstract = {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).},

author = {Omnes, Pascal},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {finite volume method; Laplace equation; Delaunay meshes; Voronoi meshes; convergence; error estimates},

language = {eng},

number = {4},

pages = {627-650},

publisher = {EDP-Sciences},

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

url = {http://eudml.org/doc/273094},

volume = {45},

year = {2011},

}

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 - Modélisation Mathématique et Analyse Numérique

PY - 2011

PB - EDP-Sciences

VL - 45

IS - 4

SP - 627

EP - 650

AB - 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).

LA - eng

KW - finite volume method; Laplace equation; Delaunay meshes; Voronoi meshes; convergence; error estimates

UR - http://eudml.org/doc/273094

ER -

## References

top- [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. Zbl0951.65080MR1618761
- [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. Zbl0951.65082MR1611742
- [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. Zbl1111.65101MR2275464
- [4] L. Angermann, Numerical solution of second-order elliptic equations on plane domains. RAIRO Modél. Math. Anal. Numér.25 (1991) 169–191. Zbl0717.65082MR1097143
- [5] R.E. Bank and D.J. Rose, Some error estimates for the box method. SIAM J. Numer. Anal.24 (1987) 777–787. Zbl0634.65105MR899703
- [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. Zbl1119.65115MR2290407
- [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 Sciences39 (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. Zbl1180.35533MR2439501
- [9] J. Breil and P.-H. Maire, A cell-centered diffusion scheme on two-dimensional unstructured meshes. J. Comput. Phys.224 (2007) 785–823. Zbl1120.65327MR2330295
- [10] Z. Cai, On the finite volume element method. Numer. Math.58 (1991) 713–735. Zbl0731.65093MR1090257
- [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. Zbl0729.65086MR1087511
- [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. Zbl1084.65112MR2139403
- [13] C. Chainais-Hillairet, Discrete duality finite volume schemes for two-dimensional drift-diffusion and energy-transport models. Internat. J. Numer. Methods Fluids59 (2009) 239–257. Zbl1154.82034MR2484267
- [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. Zbl1029.65123MR1980190
- [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. Zbl0937.65116MR1713235
- [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). MR2500950
- [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. Rouge7 (1973) 33–75. Zbl0302.65087MR343661
- [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. Zbl1152.65110MR2318807
- [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. Zbl1086.65108MR2195910
- [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. Zbl0961.76050MR1752414
- [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. Zbl1036.65084MR1897941
- [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. Zbl0981.65095MR1804748
- [23] P.A. Forsyth and P.H. Sammon, Quadratic convergence for cell-centered grids. Appl. Numer. Math.4 (1988) 377–394. Zbl0651.65086MR948505
- [24] W. Hackbucsh, On first and second order box schemes. Computing41 (1989) 277–296. Zbl0649.65052
- [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. Zbl0822.65085MR1316144
- [26] F. Hermeline, A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys.160 (2000) 481–499. Zbl0949.65101MR1763823
- [27] F. Hermeline, Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Comput. Methods Appl. Mech. Eng.192 (2003) 1939–1959. Zbl1037.65118MR1980752
- [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. Zbl0847.65075MR1377242
- [29] C. Le Potier, Finite volume scheme for highly anisotropic diffusion operators on unstructured meshes. C. R. Math. Acad. Sci. Paris340 (2005) 921–926. Zbl1076.76049MR2152280
- [30] C. Le Potier, Finite volume monotone scheme for highly anisotropic diffusion operators on unstructured triangular meshes. C. R. Math. Acad. Sci. Paris341 (2005) 787–792. Zbl1081.65086MR2188878
- [31] C. Le Potier, A nonlinear finite volume scheme satisfying maximum and minimum principles for diffusion operators. International Journal on Finite Volumes 6 (2009). MR2519614
- [32] I.D. Mishev, Finite volume methods on Voronoi meshes. Numer. Methods Partial Differ. Equ.14 (1998) 193–212. Zbl0903.65083MR1605410
- [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). MR2415169
- [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). MR2500951
- [35] E. Süli, Convergence of finite volume schemes for Poisson's equation on nonuniform meshes. SIAM J. Numer. Anal.28 (1991) 1419–1430. Zbl0802.65104MR1119276
- [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. Zbl0903.65084MR1605414
- [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. Zbl0644.65062MR933730

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