An Upwind Finite Volume Element Method for Nonlinear Convection Diffusion Problem

DOI: 10.4236/ajcm.2011.14032   PDF   HTML     4,344 Downloads   8,557 Views   Citations


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.

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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.

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The authors declare no conflicts of interest.


[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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