A second-order finite volume element method on quadrilateral meshes for elliptic equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 6, p. 1053-1067
In this paper, by use of affine biquadratic elements, we construct and analyze a finite volume element scheme for elliptic equations on quadrilateral meshes. The scheme is shown to be of second-order in ${H}^{1}$-norm, provided that each quadrilateral in partition is almost a parallelogram. Numerical experiments are presented to confirm the usefulness and efficiency of the method.
DOI : https://doi.org/10.1051/m2an:2007002
Classification:  65N30,  65N15
Keywords: finite volume element, second-order, quadrilateral meshes, error estimates
@article{M2AN_2006__40_6_1053_0,
author = {Yang, Min},
title = {A second-order finite volume element method on quadrilateral meshes for elliptic equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {40},
number = {6},
year = {2006},
pages = {1053-1067},
doi = {10.1051/m2an:2007002},
zbl = {1141.65081},
mrnumber = {2297104},
language = {en},
url = {http://www.numdam.org/item/M2AN_2006__40_6_1053_0}
}

Yang, Min. A second-order finite volume element method on quadrilateral meshes for elliptic equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 6, pp. 1053-1067. doi : 10.1051/m2an:2007002. http://www.numdam.org/item/M2AN_2006__40_6_1053_0/

[1] R.E. Bank and D.J. Rose, Some error estimates for the box method. SIAM J. Numer. Anal. 24 (1987) 777-787. | Zbl 0634.65105

[2] B. Bialecki, M. Ganesh and K. Mustapha, A Petrov-Galerkin method with quadrature for elliptic boundary value problems. IMA J. Numer. Anal. 24 (2004) 157-177. | Zbl 1057.65080

[3] Z. Cai, On the finite volume element method. Numer. Math. 58 (1991) 713-735. | Zbl 0731.65093

[4] 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. | Zbl 0729.65086

[5] S.H. Chou and S. He, On the regularity and uniformness conditions on quadrilateral grids. Comput. Methods Appl. Mech. Engrg., 191 (2002) 5149-5158. | Zbl 1030.65124

[6] S.H. Chou, D.Y. Kwak and K.Y. Kim, Mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems. Math. Comp. 72 (2002) 525-539. | Zbl 1015.65068

[7] S.H. Chou, D.Y. Kwak and Q. Li, ${L}^{p}$ error estimates and superconvergence for covolume or finite volume element methods. Num. Meth. P. D. E. 19 (2003) 463-486. | Zbl 1029.65123

[8] P.G. Ciarlett, The finite element methods for elliptic problems. North-Holland, Amsterdam, New York, Oxford (1980). | Zbl 0511.65078

[9] R.E. Ewing, R. Lazarov and Y. Lin, Finite volume element approximations of nonlocal reactive flows in porous media. Num. Meth. P. D. E. 16 (2000) 285-311. | Zbl 0961.76050

[10] 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 (2001) 1865-1888. | Zbl 1036.65084

[11] W. Hackbusch, On first and second order box schemes. Computing 41 (1989) 277-296. | Zbl 0649.65052

[12] R.E. Lynch, J.R. Rice and D.H. Thomas, Direct solution of partitial difference equations by tensor product methods. Numer. Math. 6 (1964) 185-199. | Zbl 0126.12703

[13] Y. Li and R. Li, Generalized difference methods on arbitrary quadrilateral networks. J. Comput. Math. 17 (1999) 653-672. | Zbl 0946.65098

[14] R. Li, Z. Chen and W. Wu, Generalized difference methods for differential equations, Numerical analysis of finite volume methods. Marcel Dekker, New York (2000). | MR 1731376 | Zbl 0940.65125

[15] F. Liebau, The finite volume element method with quadratic basis functions. Computing 57 (1996) 281-299. | Zbl 0866.65074

[16] I.D. Mishev, Finite volume element methods for non-definite problems. Numer. Math. 83 (1999) 161-175. | Zbl 0938.65131

[17] E. Süli, Convergence of finite volume schemes for Poisson's equation on nonuniform meshes. SIAM J. Numer. Anal. 28 (1991) 1419-1430. | Zbl 0802.65104

[18] E. Süli, The accuracy of cell vertex finite volume methods on quadrilateral meshes. Math. Comp. 59 (1992) 359-382. | Zbl 0767.65072

[19] M. Tian and Z. Chen, Generalized difference methods for second order elliptic partial differential equations. Numer. Math. J. Chinese Universities 13 (1991) 99-113. | Zbl 0734.65083

[20] Z.J. Wang, Spectral (finite) volume methods for conservation laws on unstructured grids: basic formulation. J. Comput. Phys. 178 (2002) 210-251. | Zbl 0997.65115

[21] Z.J. Wang, L. Zhang and Y. Liu, Spectral (finite) volume method for conservation laws on unstructured grids. IV: Extension to two-dimensional systems. J. Comput. Phys. 194 (2004) 716-741. | Zbl 1039.65072

[22] X. Xiang, Generalized difference methods for second order elliptic equations. Numer. Math. J. Chinese Universities 2 (1983) 114-126. | Zbl 0572.65079

[23] M. Yang and Y. Yuan, A multistep finite volume element scheme along characteristics for nonlinear convection diffusion problems. Math. Numer. Sinica 24 (2004) 487-500.