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A second-order finite volume element methodon quadrilateralmeshes for elliptic equations

Published online by Cambridge University Press:  15 February 2007

Min Yang
Min Yang*
Affiliation:
Department of Mathematics, Yantai University, Yantai, 264005, P.R. China. yangmin_math@163.com
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Abstract

In this paper, by use of affine biquadratic elements, we constructand analyze a finite volume element scheme for elliptic equations onquadrilateral meshes. The scheme is shown to be of second-order inH 1-norm, provided that each quadrilateral in partition is almosta parallelogram. Numerical experiments are presented to confirm theusefulness and efficiency of the method.

Keywords

Finite volume elementsecond-orderquadrilateral mesheserror estimates.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 6 , November 2006, pp. 1053 - 1067
Copyright
© EDP Sciences, SMAI, 2007

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References

Bank, R.E. and Rose, D.J., Some error estimates for the box method. SIAM J. Numer. Anal. 24 (1987) 777–787. CrossRef
Bialecki, B., Ganesh, M. and Mustapha, K., Petrov-Galerkin, A method with quadrature for elliptic boundary value problems. IMA J. Numer. Anal. 24 (2004) 157–177. 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
S.H. Chou and S. He, On the regularity and uniformness conditions on quadrilateral grids. Comput. Methods Appl. Mech. Engrg., 191 (2002) 5149–5158.
Chou, S.H., Kwak, D.Y. and Kim, K.Y., Mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems. Math. Comp. 72 (2002) 525–539. CrossRef
Chou, S.H., Kwak, D.Y. and Li, Q., Lp error estimates and superconvergence for covolume or finite volume element methods. Num. Meth. P. D. E. 19 (2003) 463–486. CrossRef
P.G. Ciarlett, The finite element methods for elliptic problems. North-Holland, Amsterdam, New York, Oxford (1980).
Ewing, R.E., Lazarov, R. and Lin, Y., Finite volume element approximations of nonlocal reactive flows in porous media. Num. Meth. P. D. E. 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 (2001) 1865–1888. CrossRef
Hackbusch, W., On first and second order box schemes. Computing 41 (1989) 277–296. CrossRef
Lynch, R.E., Rice, J.R. and Thomas, D.H., Direct solution of partitial difference equations by tensor product methods. Numer. Math. 6 (1964) 185–199. CrossRef
Li, Y. and Generalized di, R. Lifference methods on arbitrary quadrilateral networks. J. Comput. Math. 17 (1999) 653–672.
R. Li, Z. Chen and W. Wu, Generalized difference methods for differential equations, Numerical analysis of finite volume methods. Marcel Dekker, New York (2000).
Liebau, F., The finite volume element method with quadratic basis functions. Computing 57 (1996) 281–299. CrossRef
Mishev, I.D., Finite volume element methods for non-definite problems. Numer. Math. 83 (1999) 161–175. CrossRef
Süli, E., Convergence of finite volume schemes for Poisson's equation on nonuniform meshes. SIAM J. Numer. Anal. 28 (1991) 1419–1430. CrossRef
Süli, E., The accuracy of cell vertex finite volume methods on quadrilateral meshes. Math. Comp. 59 (1992) 359–382. CrossRef
Tian, M. and Chen, Z., Generalized difference methods for second order elliptic partial differential equations. Numer. Math. J. Chinese Universities 13 (1991) 99–113.
Wang, Z.J., Spectral (finite) volume methods for conservation laws on unstructured grids: basic formulation. J. Comput. Phys. 178 (2002) 210–251. CrossRef
Wang, Z.J., Zhang, L. and Liu, Y., Spectral (finite) volume method for conservation laws on unstructured grids. IV: Extension to two-dimensional systems. J. Comput. Phys. 194 (2004) 716–741. CrossRef
Xiang, X., Generalized difference methods for second order elliptic equations. Numer. Math. J. Chinese Universities 2 (1983) 114–126.
Yang, M. and Yuan, Y., A multistep finite volume element scheme along characteristics for nonlinear convection diffusion problems. Math. Numer. Sinica 24 (2004) 487–500.