Numerical approximation of a one-dimensional space fractional advection-dispersion equation with boundary layer

Finite element computations for singularly perturbed convection-diffusion equations have long been an attractive theme for numerical analysis. In this article, we consider the singularly perturbed fractional advection-dispersion equation (FADE) with boundary layer behavior. We derive a theoretical estimate which shows that the under-resolved case corresponds to @e

[1]  Rina Schumer,et al.  Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests , 2001 .

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

[3]  Rüdiger Verführt,et al.  A review of a posteriori error estimation and adaptive mesh-refinement techniques , 1996, Advances in numerical mathematics.

[4]  J. Maubach,et al.  Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems , 1997 .

[5]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[6]  P. Grisvard Singularities in Boundary Value Problems , 1992 .

[7]  Vickie E. Lynch,et al.  Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model , 2001 .

[8]  J. P. Roop Variational Solution of the Fractional Advection Dispersion Equation , 2004 .

[9]  V. Ervin,et al.  Variational formulation for the stationary fractional advection dispersion equation , 2006 .

[10]  Norbert Heuer,et al.  Numerical Approximation of a Time Dependent, Nonlinear, Space-Fractional Diffusion Equation , 2007, SIAM J. Numer. Anal..

[11]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[12]  D. Benson,et al.  The fractional‐order governing equation of Lévy Motion , 2000 .

[13]  W. R. Waldrop,et al.  Transport of tritium and four organic compounds during a natural-gradient experiment (MADE-2) , 1993 .

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

[15]  V. Ervin,et al.  Variational solution of fractional advection dispersion equations on bounded domains in ℝd , 2007 .

[16]  Stevens,et al.  Self-similar transport in incomplete chaos. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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

[18]  Roger Temam,et al.  New Approximation Algorithms for a Class of Partial Differential Equations Displayinging Boundary Layer Behavior , 2000 .

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

[20]  W. Woyczynski,et al.  Fractal Burgers Equations , 1998 .

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

[22]  Roger Temam,et al.  Numerical approximation of one-dimensional stationary diffusion equations with boundary layers , 2002 .

[23]  Norbert Heuer,et al.  Numerical Approximation of a Time Dependent, Non-linear, Fractional Order Diffusion Equation∗ , 2005 .

[24]  Chang-Yeol Jung,et al.  Numerical approximation of two‐dimensional convection‐diffusion equations with boundary layers , 2005 .

[25]  Fred J. Molz,et al.  A physical interpretation for the fractional derivative in Levy diffusion , 2002, Appl. Math. Lett..

[26]  A. Chaves,et al.  A fractional diffusion equation to describe Lévy flights , 1998 .