# zbMATH — the first resource for mathematics

Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in $$\mathbb R^2$$. (English) Zbl 1092.65122
Summary: We investigate the numerical approximation of the variational solution to the fractional advection dispersion equation (FADE) on bounded domains in $$\mathbb R^2$$. More specifically, we investigate the computational aspects of the Galerkin approximation using continuous piecewise polynomial basis functions on a regular triangulation of the domain. The computational challenges of approximating the solution to fractional differential equations using the finite element method stem from the fact that a fractional differential operator is a nonlocal operator. Several numerical examples are given which demonstrate approximations to FADEs.

##### MSC:
 65R20 Numerical methods for integral equations 45K05 Integro-partial differential equations 26A33 Fractional derivatives and integrals
Full Text:
##### References:
 [1] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101 [2] Benson, D.A.; Schumer, R.; Meerschaeert, M.M.; Wheatcraft, S.W., Fractional dispersion, Lévy motion and the made tracer tests, Transp. porous media, 42, 211-240, (2001) [3] Benson, D.A.; Wheatcraft, S.W.; Meerschaeert, M.M., The fractional order governing equations of Lévy motion, Water resour. res., 36, 1413-1423, (2000) [4] J.M. Boggs, L.M. Beard, W.R. Waldrop, Transport of tritium and four organic compounds during a natural-gradient experiment (made-2), Tech. Report EPRI TR-101998, Electric Power Research Institute, Pleasant Hill, California, 1993. [5] Boggs, J.M.; Young, S.C.; Beard, L.M., Field study of dispersion in a heterogeneous aquifer, 1, overview and site description, Water resour. res., 28, 3281-3291, (1992) [6] Brenner, S.; Scott, L.R., The mathematical theory of finite element methods, (1994), Springer New York · Zbl 0804.65101 [7] Carreras, B.A.; Lynch, V.E.; Zaslavsky, G.M., Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence models, Phys. plasmas, 8, 12, 5096-5103, (2001) [8] V.J. Ervin, J.P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Meth. P.D.E., 2004, accepted for publication. · Zbl 1095.65118 [9] V.J. Ervin, J.P. Roop, Variational solution of the fractional advection dispersion equation on bounded domains in $$\mathbb{R}^d$$, Numer. Meth. P.D.E., 2005, submitted for publication. [10] Fix, G.J.; Roop, J.P., Least squares finite element solution of a fractional order two-point boundary value problem, Comput. math. appl., 48, 1017-1033, (2004) · Zbl 1069.65094 [11] Liu, F.; Ahn, V.; Turner, I., Numerical solution of the space fractional fokker – planck equation, J. comp. appl. math., 166, (2004) · Zbl 1036.82019 [12] Lu, S.; Molz, F.J.; Fix, G.J., Possible problems of scale dependency in applications of the three-dimensional fractional advection – dispersion equation to natural porous media, Water resour. res., 38, 1165-1171, (2002) [13] Meerschaert, M.M.; Benson, D.A.; Baeumer, B., Multidimensional advection and fractional dispersion, Phys. rev. E, 59, 5, 5026-5028, (1999) [14] Meerschaert, M.M.; Mortenson, J., Vector Grünwald formula for fractional derivatives, Fract. calc. appl. anal., 7, 61-81, (2004) · Zbl 1084.65024 [15] M.M. Meerschaert, H.P. Scheffler, C. Tadjeran, Finite difference methods for two-dimensional fractional dispersion equations, J. Comp. Phys., to appear. · Zbl 1085.65080 [16] Meerschaert, M.M.; Tadjeran, C., Finite difference approximations for fractional advection – dispersion flow equations, J. comp. appl. math, 172, 65-77, (2004) · Zbl 1126.76346 [17] M.M. Meerschaert, C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., to appear. · Zbl 1086.65087 [18] Renardy, M.; Rogers, R., An introduction to partial differential equations, (1993), Springer New York · Zbl 0917.35001 [19] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications, (1993), Gordon & Breach New York · Zbl 0818.26003 [20] Samorodnitsky, G.; Taqqu, M.S., Stable non-Gaussian random processes: stochastic models with infinite variance, (1994), Chapman & Hall New York · Zbl 0925.60027 [21] S. Shen, F. Liu, Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends, Proceedings of the 12th Biennial Computational Techniques and Applications Conference, The University of Melbourne, Victoria, Australia. · Zbl 1078.65563 [22] Shlesinger, M.F.; West, B.J.; Klafter, J., Lévy dynamics of enhanced diffusion: application to turbulence, Phys. rev. lett., 58, 11, 1100-1103, (1987) [23] Zaslavsky, G.M.; Stevens, D.; Weitzner, H., Self-similar transport in incomplete chaos, Phys. rev. E, 48, 3, 1683-1694, (1993)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.