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Finite Element Method for a Kind of Two-Dimensional Space-Fractional Diffusion Equation with Its Implementation

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DOI: 10.4236/ajcm.2015.52012    2,762 Downloads   3,584 Views   Citations

ABSTRACT

In this article, we consider a two-dimensional symmetric space-fractional diffusion equation in which the space fractional derivatives are defined in Riesz potential sense. The well-posed feature is guaranteed by energy inequality. To solve the diffusion equation, a fully discrete form is established by employing Crank-Nicolson technique in time and Galerkin finite element method in space. The stability and convergence are proved and the stiffness matrix is given analytically. Three numerical examples are given to confirm our theoretical analysis in which we find that even with the same initial condition, the classical and fractional diffusion equations perform differently but tend to be uniform diffusion at last.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Duan, B. , Zheng, Z. and Cao, W. (2015) Finite Element Method for a Kind of Two-Dimensional Space-Fractional Diffusion Equation with Its Implementation. American Journal of Computational Mathematics, 5, 135-157. doi: 10.4236/ajcm.2015.52012.

References

[1] Metzler, R. and Klafter, J. (2004) The Restaurant at the End of the Random Walk: Recent Developments in the Description of Anomalous Transport by Fractional Dynamics. Journal of Physics A: Mathematical and General, 37, R161- R208.
http://dx.doi.org/10.1088/0305-4470/37/31/R01
[2] Zaslavsky, G.M., Stevens, D. and Weitzner, H. (1993) Self-Similar Transport in Incomplete Chaos. Physical Review E, 48, 1683-1694.
http://dx.doi.org/10.1103/PhysRevE.48.1683
[3] Metzler, R. and Klafter, J. (2000) The Random Walk’s Guide to Anomalous Diffusion: A Fractional Dynamics Approach. Physics Reports, 339, 1-77.
http://dx.doi.org/10.1016/S0370-1573(00)00070-3
[4] Zaslavsky, G.M. (2002) Chaos, Fractional Kinetics, and Anomalous Transport. Physics Reports, 371, 461-580.
http://dx.doi.org/10.1016/S0370-1573(02)00331-9
[5] Schumer, R., Benson, D.A., Meerschaert, M.M. and Baeumer, B. (2003) Multiscaling Fractional Advection-Dispersion Equations and Their Solutions. Water Resources Research, 39, 1022-1032.
http://dx.doi.org/10.1029/2001WR001229
[6] Schumer, R., Benson, D.A., Meerschaert, M.M. and Wheatcraft, S.W. (2001) Eulerian Derivation of the Fractional Advection-Dispersion Equation. Journal of Contaminant Hydrology, 48, 69-88.
http://dx.doi.org/10.1016/S0169-7722(00)00170-4
[7] Tadjeran, C., Meerschaert, M.M. and Scheffler, H.P. (2006) A Second-Order Accurate Numerical Approximation for the Fractional Diffusion Equation. Journal of Computational Physics, 213, 205-213.
http://dx.doi.org/10.1016/j.jcp.2005.08.008
[8] Tadjeran, C. and Meerschaert, M.M. (2007) A Second-Order Accurate Numerical Method for the Two-Dimensional Fractional Diffusion Equation. Journal of Computational Physics, 220, 813-823.
http://dx.doi.org/10.1016/j.jcp.2006.05.030
[9] Sousa, E. (2011) Numerical Approximations for Fractional Diffusion Equations via Splines. Computers and Mathematics with Applications, 62, 938-944.
http://dx.doi.org/10.1016/j.camwa.2011.04.015
[10] Li, X. and Xu, C. (2009) A Space-Time Spectral Method for the Time Fractional Diffusion Equation. SIAM Journal on Numerical Analysis, 47, 2108-2131.
[11] Xu, Q. and Hesthaven, J.S. (2013) Discontinuous Galerkin Method for Fractional Convection-Diffusion Equations.
http://arxiv.org/abs/1304.6047
[12] Jin, B., Lazarov, R., Liu, Y. and Zhou, Z. (2015) The Galerkin Finite Element Method for a Multi-Term Time-Fractional Diffusion Equation. Journal of Computational Physics, 281, 825-843.
http://dx.doi.org/10.1016/j.jcp.2014.10.051
[13] Chaves, A.S. (1998) A Fractional Diffusion Equation to Describe Lévy Flight. Physics Letters A, 239, 13-16.
http://dx.doi.org/10.1016/S0375-9601(97)00947-X
[14] Benson, D.A., Wheatcraft, S.W. and Meerschaert, M.M. (2000) Application of a Fractional Advection-Dispersion Equation. Water Resources Research, 36, 1403-1412.
http://dx.doi.org/10.1029/2000WR900031
[15] Benson, D.A., Wheatcraft, S.W. and Meerschaert, M.M. (2000) The Fractional Order Governing Equation of Lévy Motion. Water Resources Research, 36, 1413-1423.
http://dx.doi.org/10.1029/2000WR900032
[16] Zhang, H., Liu, F. and Anh, V. (2010) Galerkin Finite Element Approximation of Symmetric Space-Fractional Partial Differential Equations. Applied Mathematics and Computation, 217, 2534-2545.
http://dx.doi.org/10.1016/j.amc.2010.07.066
[17] Sousa, E. (2012) A Second Order Explicit Finite Difference Method for the Fractional Advection Diffusion Equation. Computers and Mathematics with Applications, 64, 3141-3152.
http://dx.doi.org/10.1016/j.camwa.2012.03.002
[18] Liu, F., Zhuang, P., Turner, I., Anh, V. and Burrage, K. (2015) A Semi-Alternating Direction Method for a 2-D Fractional FitzHugh-Nagumo Monodomain Model on an Approximate Irregular Domain. Journal of Computational Physics, 293, 252-263.
[19] Zeng, F., Liu, F., Li, C., Burrage, K., Turner, I. and Anh, V. (2014) Crank-Nicolson ADI Spectral Method for the Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation. SIAM Journal on Numerical Analysis, 52, 2599-2622.
http://dx.doi.org/10.1137/130934192
[20] Wang, H. and Du, N. (2013) A Fast Finite Difference Method for Three-Dimensional Time-Dependent Space-Fractional Diffusion Equations and Its Efficient Implementation. Journal of Computational Physics, 253, 50-63.
http://dx.doi.org/10.1016/j.jcp.2013.06.040
[21] Wang, H. and Du, N. (2014) Fast Alternating-Direction Finite Difference Methods for Three-Dimensional Space-Fractional Diffusion Equations. Journal of Computational Physics, 258, 305-318.
http://dx.doi.org/10.1016/j.jcp.2013.10.040
[22] Ervin, V.J. and Roop, J.P. (2006) Variational Formulation for the Stationary Fractional Advection Dispersion Equation. Numerical Methods for Partial Differential Equations, 22, 558-576.
http://dx.doi.org/10.1002/num.20112
[23] Ervin, V.J. and Roop, J.P. (2007) Variational Solution of Fractional Advection Dispersion Equations on Bounded Domains in Rd. Numerical Methods for partial Differential Equations, 23, 256-281.
http://dx.doi.org/10.1002/num.20169
[24] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, Waltham.
[25] Muslih, S.I. and Agrawal, O.P. (2010) Riesz Fractional Derivatives and Fractional Dimensional Space. International Journal of Theoretical Physics, 49, 270-275.
http://dx.doi.org/10.1007/s10773-009-0200-1
[26] El-Sayed, A.M.A. and Gaber, M. (2006) On the Finite Caputo and Finite Riesz Derivatives. Electronic Journal of Theoretical Physics, 3, 81-95.
[27] Brenner, S. and Scott, L.R. (1994) The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4757-4338-8
[28] Roop, J.P. (2006) Computational Aspects of FEM Approximation of Fractional Advection Dispersion Equations on Bounded Domains in R2. Journal of Computational and Applied Mathematics, 193, 243-268.
[29] Metzler, R. and Klafter, J. (2004) The Restaurant at the End of the Random Walk: Recent Developments in the Description of Anomalous Transport by Fractional Dynamics. Journal of Physics A: Mathematical and General, 37, R161- R208.
http://dx.doi.org/10.1088/0305-4470/37/31/R01
[30] Zaslavsky, G.M., Stevens, D. and Weitzner, H. (1993) Self-Similar Transport in Incomplete Chaos. Physical Review E, 48, 1683-1694.
http://dx.doi.org/10.1103/PhysRevE.48.1683
[31] Metzler, R. and Klafter, J. (2000) The Random Walk’s Guide to Anomalous Diffusion: A Fractional Dynamics Approach. Physics Reports, 339, 1-77.
http://dx.doi.org/10.1016/S0370-1573(00)00070-3
[32] Zaslavsky, G.M. (2002) Chaos, Fractional Kinetics, and Anomalous Transport. Physics Reports, 371, 461-580.
http://dx.doi.org/10.1016/S0370-1573(02)00331-9
[33] Schumer, R., Benson, D.A., Meerschaert, M.M. and Baeumer, B. (2003) Multiscaling Fractional Advection-Dispersion Equations and Their Solutions. Water Resources Research, 39, 1022-1032.
http://dx.doi.org/10.1029/2001WR001229
[34] Schumer, R., Benson, D.A., Meerschaert, M.M. and Wheatcraft, S.W. (2001) Eulerian Derivation of the Fractional Advection-Dispersion Equation. Journal of Contaminant Hydrology, 48, 69-88.
http://dx.doi.org/10.1016/S0169-7722(00)00170-4
[35] Tadjeran, C., Meerschaert, M.M. and Scheffler, H.P. (2006) A Second-Order Accurate Numerical Approximation for the Fractional Diffusion Equation. Journal of Computational Physics, 213, 205-213.
http://dx.doi.org/10.1016/j.jcp.2005.08.008
[36] Tadjeran, C. and Meerschaert, M.M. (2007) A Second-Order Accurate Numerical Method for the Two-Dimensional Fractional Diffusion Equation. Journal of Computational Physics, 220, 813-823.
http://dx.doi.org/10.1016/j.jcp.2006.05.030
[37] Sousa, E. (2011) Numerical Approximations for Fractional Diffusion Equations via Splines. Computers and Mathematics with Applications, 62, 938-944.
http://dx.doi.org/10.1016/j.camwa.2011.04.015
[38] Li, X. and Xu, C. (2009) A Space-Time Spectral Method for the Time Fractional Diffusion Equation. SIAM Journal on Numerical Analysis, 47, 2108-2131.
[39] Xu, Q. and Hesthaven, J.S. (2013) Discontinuous Galerkin Method for Fractional Convection-Diffusion Equations.
http://arxiv.org/abs/1304.6047
[40] Jin, B., Lazarov, R., Liu, Y. and Zhou, Z. (2015) The Galerkin Finite Element Method for a Multi-Term Time-Fractional Diffusion Equation. Journal of Computational Physics, 281, 825-843.
http://dx.doi.org/10.1016/j.jcp.2014.10.051
[41] Chaves, A.S. (1998) A Fractional Diffusion Equation to Describe Lévy Flight. Physics Letters A, 239, 13-16.
http://dx.doi.org/10.1016/S0375-9601(97)00947-X
[42] Benson, D.A., Wheatcraft, S.W. and Meerschaert, M.M. (2000) Application of a Fractional Advection-Dispersion Equation. Water Resources Research, 36, 1403-1412.
http://dx.doi.org/10.1029/2000WR900031
[43] Benson, D.A., Wheatcraft, S.W. and Meerschaert, M.M. (2000) The Fractional Order Governing Equation of Lévy Motion. Water Resources Research, 36, 1413-1423.
http://dx.doi.org/10.1029/2000WR900032
[44] Zhang, H., Liu, F. and Anh, V. (2010) Galerkin Finite Element Approximation of Symmetric Space-Fractional Partial Differential Equations. Applied Mathematics and Computation, 217, 2534-2545.
http://dx.doi.org/10.1016/j.amc.2010.07.066
[45] Sousa, E. (2012) A Second Order Explicit Finite Difference Method for the Fractional Advection Diffusion Equation. Computers and Mathematics with Applications, 64, 3141-3152.
http://dx.doi.org/10.1016/j.camwa.2012.03.002
[46] Liu, F., Zhuang, P., Turner, I., Anh, V. and Burrage, K. (2015) A Semi-Alternating Direction Method for a 2-D Fractional FitzHugh-Nagumo Monodomain Model on an Approximate Irregular Domain. Journal of Computational Physics, 293, 252-263.
[47] Zeng, F., Liu, F., Li, C., Burrage, K., Turner, I. and Anh, V. (2014) Crank-Nicolson ADI Spectral Method for the Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation. SIAM Journal on Numerical Analysis, 52, 2599-2622.
http://dx.doi.org/10.1137/130934192
[48] Wang, H. and Du, N. (2013) A Fast Finite Difference Method for Three-Dimensional Time-Dependent Space-Fractional Diffusion Equations and Its Efficient Implementation. Journal of Computational Physics, 253, 50-63.
http://dx.doi.org/10.1016/j.jcp.2013.06.040
[49] Wang, H. and Du, N. (2014) Fast Alternating-Direction Finite Difference Methods for Three-Dimensional Space-Fractional Diffusion Equations. Journal of Computational Physics, 258, 305-318.
http://dx.doi.org/10.1016/j.jcp.2013.10.040
[50] Ervin, V.J. and Roop, J.P. (2006) Variational Formulation for the Stationary Fractional Advection Dispersion Equation. Numerical Methods for Partial Differential Equations, 22, 558-576.
http://dx.doi.org/10.1002/num.20112
[51] Ervin, V.J. and Roop, J.P. (2007) Variational Solution of Fractional Advection Dispersion Equations on Bounded Domains in Rd. Numerical Methods for partial Differential Equations, 23, 256-281.
http://dx.doi.org/10.1002/num.20169
[52] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, Waltham.
[53] Muslih, S.I. and Agrawal, O.P. (2010) Riesz Fractional Derivatives and Fractional Dimensional Space. International Journal of Theoretical Physics, 49, 270-275.
http://dx.doi.org/10.1007/s10773-009-0200-1
[54] El-Sayed, A.M.A. and Gaber, M. (2006) On the Finite Caputo and Finite Riesz Derivatives. Electronic Journal of Theoretical Physics, 3, 81-95.
[55] Brenner, S. and Scott, L.R. (1994) The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4757-4338-8
[56] Roop, J.P. (2006) Computational Aspects of FEM Approximation of Fractional Advection Dispersion Equations on Bounded Domains in R2. Journal of Computational and Applied Mathematics, 193, 243-268.

  
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