A second-order finite volume element method on quadrilateral meshes for elliptic equations
ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 6, pp. 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 H1-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 : 10.1051/m2an:2007002
Classification : 65N30, 65N15
Mots-clés : finite volume element, second-order, quadrilateral meshes, error estimates
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     title = {A second-order finite volume element method on quadrilateral meshes for elliptic equations},
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Yang, Min. A second-order finite volume element method on quadrilateral meshes for elliptic equations. ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 6, pp. 1053-1067. doi : 10.1051/m2an:2007002. http://www.numdam.org/articles/10.1051/m2an:2007002/

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

[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

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

[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

[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

[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

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

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

[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

[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

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

[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

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

[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 | Zbl

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

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

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

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

[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

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

[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

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

[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.

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