Finite element method for two-dimensional space-fractional advection-dispersion equations

The backward Euler and Crank-Nicolson-Galerkin fully-discrete approximate schemes for two-dimensional space-fractional advection-dispersion equations are established. Firstly, we prove that the corresponding variational problem has a unique solution, and the proposed fully-discrete schemes are unconditionally stable, whose solutions are all unique. Secondly, the optimal error estimates are derived by use of properties of projection operator and fractional derivatives. Finally, numerical examples demonstrate effectiveness of numerical schemes and confirm the theoretical analysis.