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An Upwind Finite Volume Element Method for Nonlinear Convection Diffusion Problem

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DOI: 10.4236/ajcm.2011.14032    4,166 Downloads   8,405 Views   Citations

ABSTRACT

A class of upwind finite volume element method based on tetrahedron partition is put forward for a nonlinear convection diffusion problem. Some techniques, such as calculus of variations, commutating operators and the a priori estimate, are adopted. The a priori error estimate in L2-norm and H1-norm is derived to determine the error between the approximate solution and the true solution.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

F. Gao, Y. Yuan and N. Du, "An Upwind Finite Volume Element Method for Nonlinear Convection Diffusion Problem," American Journal of Computational Mathematics, Vol. 1 No. 4, 2011, pp. 264-270. doi: 10.4236/ajcm.2011.14032.

References

[1] R. H. Li, Z. Y. Chen and W. Wu, “Generalized Differ- ence Methods for Differential Equations: Numerical Ana- lysis of Finite Volume Methods,” Marcel Dekker, New York, 2000.
[2] Z. Q, Cai and S. F. McCormick, “On the Accuracy of the Finite Volume Element Method for Diffusion Equations on Composite Grids,” SIAM Journal on Numerical Analysis, Vol. 27, 1990, pp. 635-655, 1990.
[3] Z. Q. Cai,J. Mandel and S. F. McCormick, “The Finite Volume Element Method for Diffusion Equations on Ge- neral Triangulations,” SIAM Journal on Numerical Ana- lysis, Vol. 28, No. 2, 1991, pp. 392-402. doi:10.1137/0728022
[4] R. E. Bank and D. J. Rose, “Some Error Estimates for the Box Method,” SIAM Journal on Numerical Analysis, Vol. 24, No. 4, 1987, pp. 777-787. doi:10.1137/0724050
[5] V. Patankar, “Numerical Heat Transfer and Fluid Flow,” McGraw-Hill, New York, 1980.
[6] J. Douglas Jr. and T. F. Russell, “Numerical Methods for Convection-Dominated Diffusion Problems Based on Com- bining the Method of Characteristics with Finite Element or Finite Difference Procedures,” SIAM Journal on Numerical Analysis, Vol. 19, No. 5, 1982, pp. 871-885. doi:10.1137/0719063
[7] D. B. Spalding, “A Novel Finite Difference Formulation for Differential Equations Involving Both First and Sec- ond Derivatives,” International Journal for Numerical Me- thods in Engineering, Vol. 4, No. 4, 1973, pp. 551-559. doi:10.1002/nme.1620040409
[8] K. Baba and M. Tabata, “On a Conservative Upwind Finite Element Scheme for Convective Diffusion Equa- tions,” RAIRO Analyse Numériqe, Vol. 15, No. 1, 1981, pp. 3-25.
[9] M. Tabata, “Uniform Convergence of the Upwind Finite Element Approximation for Semi-Linear Parabolic Prob- lems,” Journal of Mathematics of Kyoto University, Vol. 18, No. 2, 1978, pp. 307-351.
[10] M. Tabata, “A Finite Element Approximation Correspon- ding to the Upwind Finite Differencing,” Memoirs of Numerical Mathematics, Vol. 4, 1977, pp. 47-63.
[11] M. Tabata, “Conservative Upwind Finite Element Appro- ximation and Its Applications, Analytical and Numerical Approaches to Asymptopic Problem in Analysis,” North- Holland, Amsterdam, 1981, pp. 369-387.
[12] Y. R. Yuan, “The Upwind Finite Difference Fractional Steps Methods for Two-phase Compressible Flow in Po- rous Media,” Numerical Methods for Partial Differential Equations, Vol. 19, No. 1, 2003, pp. 67-88. doi:10.1002/num.10036
[13] D. Liang, “A Kind of Upwind Schemes for Convection Diffusion Equations,” Math. Numer. Sinica, Vol. 2, 1991, pp. 133-141.
[14] Y. H. Li and R. H. Li, “Generalized Difference Methods on Arbitrary Quadrilateral Netwoks,” Journal of Compu- tational Mathematics, Vol. 17, No. 6, 1999, pp. 653-672.
[15] M. Feistauer, J. Felcman and M. Luká?ová-Medvid'ová, “On the Convergence of a Combined Finite Volume-Finite Element Method for Nonlinear Convection-Diffusion Pro- blems,” Numerical Methods for Partial Differential Equa- tions, Vol. 13, No. 2, 1997, pp. 163-190. doi:10.1002/(SICI)1098-2426(199703)13:2<163::AID-NUM3>3.0.CO;2-N
[16] M. Feistauer, J. Slavik and P. Stupka, “On the Conver- gence of a Combined Finite Volume-Finite Element Me- thod for Nonlinear Convection-Diffusion Problems,” Nu- merical Methods for Partial Differential Equations, Vol. 15, No. 2, 1999, pp. 215-235. doi:10.1002/(SICI)1098-2426(199903)15:2<215::AID-NUM6>3.0.CO;2-1
[17] F. Z. Gao and Y. R. Yuan. “An upwind Finite Volume Element Method Based on Quadrilateral Meshes for Nonlinear Convection-Diffusion Problems,” Numerical Methods for Partial Differential Equations, Vol. 25, No. 5, 2009, pp. 1067-1085. doi:10.1002/num.20387
[18] P. G. Ciarlet, “The Finite Element Method for Elliptic Pro- blems,” North-Holland, Amsterdam, 1978.
[19] R. A. Adams, “Sobolev Spaces,” Academic Press, New York, 1975.
[20] S. C. Brenner and L. R. Scott, “The Mathematics Theory of Finite Element Methods,” Springer-Verlag, New York, 1994.

  
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