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A SYMMETRIC FINITE VOLUME ELEMENT SCHEME ON TETRAHEDRON GRIDS
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 Title & Authors
A SYMMETRIC FINITE VOLUME ELEMENT SCHEME ON TETRAHEDRON GRIDS
Nie, Cunyun; Tan, Min;
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 Abstract
We construct a symmetric finite volume element (SFVE) scheme for a self-adjoint elliptic problem on tetrahedron grids and prove that our new scheme has optimal convergent order for the solution and has superconvergent order for the flux when grids are quasi-uniform and regular. The symmetry of our scheme is helpful to solve efficiently the corresponding discrete system. Numerical experiments are carried out to confirm the theoretical results.
 Keywords
symmetry;finite volume element scheme;superconvergence;tetrahedron grids;
 Language
English
 Cited by
 References
1.
A. A. Abedini and R. A. Ghiassi, A three-dimensional finite volume model for shallow water flow simulation, Ausrtalian Journal of Basic AS. 4 (2010), 3208-3215.

2.
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

3.
H. Baek, S. D. Kim, and H. C. Lee, A multigrid method for an optimal control problem of a diffusion-convection equation, J. Korean Math. Soc. 47 (2010), no. 1, 83-100. crossref(new window)

4.
R. E. Bank and D. J. Rose, Some error estimates for the box method, SIAM J. Numer. Anal. 24 (1987), no. 4, 777-787. crossref(new window)

5.
M. Benjemaa, N. Glinsky-Olivier, V. M. Cruz-Atienza, and J. Virieux, 3-D dynamic rupture simulations by a finite volume method, Int. J. Geophys. 178 (2009), 541-560. crossref(new window)

6.
Z. Cai, On the finite volume element method, Numer. Math. 58 (1991), no. 7, 713-735.

7.
L. Chen, Superconvergence of tetrahedral finite elements, Inter. J. Numer. Anal. Model. 3 (2006), 273-282.

8.
C. M. Chen and Y. Huang, High accuarcy theory of finite element methods, Science Press, Hunan, China (in Chinese), 1995.

9.
S. Chou and Q. Li, Error estimates in $L^p,\;L^{\infty}\;and\;H^1$ in covolume methods for ellipitic and parabolic problems, Math. Comp. 69 (1999), 103-120. crossref(new window)

10.
P. G. Ciarlet, The Finite Element Method for Elliptie Problems, North-Holland, Amsterdam, New York, Oxford, 1978.

11.
R. Ewing, R. Lazarov, and Y. Lin, Finite volume element approximations of nonlocal reactive flows in porous media, Numer. Methods Partial Differential Equations 16 (2000), no. 3, 285-311. crossref(new window)

12.
G. Goodsell, Pointwise superconvergence of the gradient for the linear tetrahedral element, Numer. Methods Partial Differential Equations 10 (1994), no. 5, 651-666. crossref(new window)

13.
C. A. Hall, T. A. Porsching, and G. L. Mesina, On a network metod for unsteady incompressible fiuid flow on triangular grids, Int. J. Numer. Meth. FL. 15 (1992), 1383-1406. crossref(new window)

14.
J. Huang and S. Xi, On the finite volume element method for general self-adjoint elliptic problems, SIAM J. Numer. Anal. 35 (1998), no. 5, 1762-1774. crossref(new window)

15.
B. Li and Z. Zhang, Analysis of a class of superconvergence patch recovery techniches for linear and bilinear finite element, Numer. Methods Partial Differential Equations 15 (1997), 151-167.

16.
S. Liang, X. Ma, and A. Zhou, A symmetric finite volume scheme for selfadjoint elliptic problems, J. Comput. Appl. Math. 147 (2002), no. 1, 121-136. crossref(new window)

17.
J. L. Lv and Y. H. Li, $L^2$ error estimate of the finite volume element methods on quadrilateral meshes, Adv. Comput. Math. 33 (2010), no. 2, 129-148. crossref(new window)

18.
X. Ma, S. Shu, and A. Zhou, Symmetric finite volume discretization for parabolic problems, Comput. Methods Appl. Mech. Engrg. 192 (2003), no. 39-40, 4467-4485. crossref(new window)

19.
N. K. Madsena and R. W. Ziolkowskia, A three-dimensional modified finite volume technique for Maxwell's equations, Electromagnetics 10 (1990), 147-161. crossref(new window)

20.
T. J. Moroney and I. W. Turner, A three-dimensional finite volume method based on radial basis functions for the accurate computational modelling of nonlinear diffusion equations, J. Comput. Phys. 225 (2007), no. 2, 1409-1426. crossref(new window)

21.
C. Y. Nie and S. Shu, Symmetry-preserving finite volume element scheme on unstructured quadrilateral grids, Chinese J. Comp. Phys. 26 (2004), 17-22.

22.
H. Rui, Symmetric modified finite volume element methods for self-adjoint elliptic and parabolic problems, J. Comput. Appl. Math. 146 (2002), no. 2, 373-386. crossref(new window)

23.
H. Rui, Convergence of an upwind control-volume mixed finite element method for convection diffusion problems, Computing 81 (2007), no. 4, 297-315. crossref(new window)

24.
S. Shu, H. Y. Yu, Y. Q. Huang, and C. Y. Nie, A preserving-symmetry finite volume scheme and superconvergence on quadrangle grids, Inter. J. Numer. Anal. Model. 3 (2006), 348-360.

25.
D. D. Sun and S. Shu, An algebraic multigrid method of the high order Lagrangian finite element equation in $R^3$, Mathematic Numerica Sinca 1 (2005), 101-112.

26.
T. Tanaka, Finite volume TVD scheme on an unstructured grid system for three-dimensional MHD simulation of inhomogeneous systems including strong backgroud potential fields, J. Compt. Phys. 3 (1994), 381-389.

27.
W. L. Wan, T. F. Chan, and B. Smith, An energy-minimizing interpolation for robust multigrid methods, SIAM J. Sci. Comput. 21 (2000), no. 4, 1632-1649.

28.
Y. X. Xiao, S. Shu, and T. Y. Zhao, A geometric-based algebraic multigrid method for higher-order finite element equations in two-dimensional linear elasticity, Numer. Linear Algebra Appl. 16 (2009), no. 7, 535-559. crossref(new window)

29.
J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581-613. crossref(new window)