# A second-order finite volume element method on quadrilateral meshes for elliptic equations

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

- Volume: 40, Issue: 6, page 1053-1067
- ISSN: 0764-583X

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topYang, Min. "A second-order finite volume element method on quadrilateral meshes for elliptic equations." ESAIM: Mathematical Modelling and Numerical Analysis 40.6 (2007): 1053-1067. <http://eudml.org/doc/194344>.

@article{Yang2007,

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.
},

author = {Yang, Min},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Finite volume element; second-order; quadrilateral meshes;
error estimates.; mixed finite element finite volume method; elliptic problem; convergence; a priori error estimates},

language = {eng},

month = {2},

number = {6},

pages = {1053-1067},

publisher = {EDP Sciences},

title = {A second-order finite volume element method on quadrilateral meshes for elliptic equations},

url = {http://eudml.org/doc/194344},

volume = {40},

year = {2007},

}

TY - JOUR

AU - Yang, Min

TI - A second-order finite volume element method on quadrilateral meshes for elliptic equations

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2007/2//

PB - EDP Sciences

VL - 40

IS - 6

SP - 1053

EP - 1067

AB -
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.

LA - eng

KW - Finite volume element; second-order; quadrilateral meshes;
error estimates.; mixed finite element finite volume method; elliptic problem; convergence; a priori error estimates

UR - http://eudml.org/doc/194344

ER -

## References

top- R.E. Bank and D.J. Rose, Some error estimates for the box method. SIAM J. Numer. Anal.24 (1987) 777–787.
- 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.
- Z. Cai, On the finite volume element method. Numer. Math.58 (1991) 713–735.
- 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.
- S.H. Chou and S. He, On the regularity and uniformness conditions on quadrilateral grids. Comput. Methods Appl. Mech. Engrg., 191 (2002) 5149–5158.
- 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.
- 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.
- P.G. Ciarlett, The finite element methods for elliptic problems. North-Holland, Amsterdam, New York, Oxford (1980).
- 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.
- 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.
- W. Hackbusch, On first and second order box schemes. Computing41 (1989) 277–296.
- 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.
- Y. Li and R. Li, Generalized difference 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).
- F. Liebau, The finite volume element method with quadratic basis functions. Computing57 (1996) 281–299.
- I.D. Mishev, Finite volume element methods for non-definite problems. Numer. Math.83 (1999) 161–175.
- E. Süli, Convergence of finite volume schemes for Poisson's equation on nonuniform meshes. SIAM J. Numer. Anal.28 (1991) 1419–1430.
- E. Süli, The accuracy of cell vertex finite volume methods on quadrilateral meshes. Math. Comp.59 (1992) 359–382.
- M. Tian and Z. Chen, Generalized difference methods for second order elliptic partial differential equations. Numer. Math. J. Chinese Universities13 (1991) 99–113.
- Z.J. Wang, Spectral (finite) volume methods for conservation laws on unstructured grids: basic formulation. J. Comput. Phys.178 (2002) 210–251.
- 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.
- X. Xiang, Generalized difference methods for second order elliptic equations. Numer. Math. J. Chinese Universities2 (1983) 114–126.
- M. Yang and Y. Yuan, A multistep finite volume element scheme along characteristics for nonlinear convection diffusion problems. Math. Numer. Sinica24 (2004) 487–500.

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