A semi-discrete finite element method for a class of time-fractional diffusion equations

As fractional diffusion equations can describe the early breakthrough and the heavy-tail decay features observed in anomalous transport of contaminants in groundwater and porous soil, they have been commonly used in the related mathematical descriptions. These models usually involve long-time-range computation, which is a critical obstacle for their application; improvement of computational efficiency is of great significance. In this paper, a semi-discrete method is presented for solving a class of time-fractional diffusion equations that overcome the critical long-time-range computation problem. In the procedure, the spatial domain is discretized by the finite element method, which reduces the fractional diffusion equations to approximate fractional relaxation equations. As analytical solutions exist for the latter equations, the burden arising from long-time-range computation can effectively be minimized. To illustrate its efficiency and simplicity, four examples are presented. In addition, the method is used to solve the time-fractional advection–diffusion equation characterizing the bromide transport process in a fractured granite aquifer. The prediction closely agrees with the experimental data, and the heavy-tail decay of the anomalous transport process is well represented.

[1]  J. P. Roop.  Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R 2 , 2006 .

[2]  D. V. Griffiths,et al.  Programming the finite element method , 1982 .

[3]  Graham E. Fogg,et al.  Role of Molecular Diffusion in Contaminant Migration and Recovery in an Alluvial Aquifer System , 2001 .

[4]  G. Zaslavsky.  Chaos, fractional kinetics, and anomalous transport , 2002 .

[5]  Fawang Liu,et al.  Galerkin finite element approximation of symmetric space-fractional partial differential equations , 2010, Appl. Math. Comput..

[6]  Hongguang Sun,et al.  Finite difference Schemes for Variable-Order Time fractional Diffusion equation , 2012, Int. J. Bifurc. Chaos.

[7]  Roland W. Lewis,et al.  The Finite Element Method in Heat Transfer Analysis , 1996 .

[8]  D. Benson,et al.  Radial fractional‐order dispersion through fractured rock , 2004 .

[9]  M. Dentz,et al.  Modeling non‐Fickian transport in geological formations as a continuous time random walk , 2006 .

[10]  D. Benson,et al.  Operator Lévy motion and multiscaling anomalous diffusion. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Changpin Li,et al.  Finite difference Methods for fractional differential equations , 2012, Int. J. Bifurc. Chaos.

[12]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[13]  K. Y. Sze,et al.  A hybrid‐Trefftz finite element model for Helmholtz problem , 2008 .

[14]  Robin Gerlach,et al.  Magnetic resonance microscopy of biofouling induced scale dependent transport in porous media , 2007 .

[15]  Brian Berkowitz,et al.  On Characterization of Anomalous Dispersion in Porous and Fractured Media , 1995 .

[16]  Fawang Liu,et al.  Novel Numerical Methods for Solving the Time-Space Fractional Diffusion Equation in Two Dimensions , 2011, SIAM J. Sci. Comput..

[17]  Xiaohong Joe Zhou,et al.  Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation. , 2008, Journal of magnetic resonance.

[18]  Guanhua Huang,et al.  A finite element solution for the fractional advection–dispersion equation , 2008 .

[19]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

[20]  K. Diethelm.  Mittag-Leffler Functions , 2010 .

[21]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[22]  F. Mainardi.  Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena , 1996 .

[23]  I. Podlubny.  Fractional differential equations , 1998 .

[24]  G. L. Guymon,et al.  A finite element solution of the one-dimensional diffusion-convection equation , 1970 .

[25]  Zhaoxia Yang,et al.  Finite difference approximations for the fractional advection-diffusion equation , 2009 .

[26]  D. Benson,et al.  Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications , 2009 .

[27]  I M Sokolov,et al.  From diffusion to anomalous diffusion: a century after Einstein's Brownian motion. , 2005, Chaos.

[28]  Neville J. Ford,et al.  The numerical solution of fractional differential equations: Speed versus accuracy , 2001, Numerical Algorithms.

[29]  Sabine Attinger,et al.  Accuracy of numerical simulations of contaminant transport in heterogeneous aquifers: A comparative study , 2011 .

[30]  Yangquan Chen,et al.  Matrix approach to discrete fractional calculus II: Partial fractional differential equations , 2008, J. Comput. Phys..

[31]  Santos B. Yuste,et al.  Weighted average finite difference methods for fractional diffusion equations , 2004, J. Comput. Phys..

[32]  G. Fix,et al.  Least squares finite-element solution of a fractional order two-point boundary value problem , 2004 .

[33]  A. Hassan,et al.  Modeling Ground Water Flow and Radioactive Transport in a Fractured Aquifer , 1999, Ground water.

[34]  John W. Crawford,et al.  The impact of boundary on the fractional advection dispersion equation for solute transport in soil: Defining the fractional dispersive flux with the Caputo derivatives , 2007 .

[35]  P. Reimus,et al.  Testing and parameterizing a conceptual solute transport model in saturated fractured tuff using sorbing and nonsorbing tracers in cross-hole tracer tests. , 2003, Journal of contaminant hydrology.

[36]  Paul W. Reimus,et al.  Testing and parameterizing a conceptual model for solute transport in a fractured granite using multiple tracers in a forced‐gradient test , 2003 .

[37]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[38]  Changpin Li,et al.  A Fully Discrete Discontinuous Galerkin Method for Nonlinear Fractional Fokker-Planck Equation , 2010 .

[39]  Y. Chen,et al.  Variable-order fractional differential operators in anomalous diffusion modeling , 2009 .

[40]  Yangquan Chen,et al.  Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion , 2011, Comput. Math. Appl..

[41]  D. Benson,et al.  Application of a fractional advection‐dispersion equation , 2000 .

[42]  Changpin Li,et al.  Numerical approaches to fractional calculus and fractional ordinary differential equation , 2011, J. Comput. Phys..

[43]  Fawang Liu,et al.  A Fourier method for the fractional diffusion equation describing sub-diffusion , 2007, J. Comput. Phys..

[44]  Weihua Deng,et al.  Finite Element Method for the Space and Time Fractional Fokker-Planck Equation , 2008, SIAM J. Numer. Anal..

[45]  Changpin Li,et al.  A note on the finite element method for the space-fractional advection diffusion equation , 2010, Comput. Math. Appl..

[46]  Arak M. Mathai,et al.  Mittag-Leffler Functions and Their Applications , 2009, J. Appl. Math..

[47]  Richard L. Magin,et al.  Solving the fractional order Bloch equation , 2009 .

[48]  Gedeon Dagan,et al.  Theory of Solute Transport by Groundwater , 1987 .

[49]  Pankaj Kumar,et al.  An approximate method for numerical solution of fractional differential equations , 2006, Signal Process..

[50]  Jean-Michel Bergheau,et al.  Finite Element Simulation of Heat Transfer , 2008 .