Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-19T18:44:34.260Z Has data issue: false hasContentIssue false
  • English
  • Français

A second-order finite volume element method on 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
Get access

Abstract

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.

Keywords

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.