A Comparative Study of Finite Element and Finite Difference Methods for Two-Dimensional Space-Fractional Advection-Dispersion Equation (Advances in Applied Mathematics and Mechanics, 2018)
The paper makes a comparative study of the finite element method (FEM)and the finite difference method (FDM) for two-dimensional fractional advection-dispersionequation (FADE) which has recently been considered a promising tool inmodeling non-Fickian solute transport in groundwater. Due to the non-local propertyof integro-differential operator of the space-fractional derivative, numerical solution ofFADE is very challenging and little has been reported in literature, especially for high-dimensionalcase. In order to effectively apply the FEM and the FDM to the FADEon a rectangular domain, a backward-distance algorithm is presented to extend thetriangular elements to generic polygon elements in the finite element analysis, and avariable-step vector Gr\u00fcnwald formula is proposed to improve the solution accuracy of the conventional finite difference scheme. Numerical investigation shows that theFEM compares favorably with the FDM in terms of accuracy and convergence ratewhereas the latter enjoys less computational effort.