  We provide a mathematical framework for studying different versions of
discontinuous Galerkin (DG) approaches for solving 2D Riemann-Liouville
fractional elliptic problems on a finite domain. The boundedness and stability
analysis of the primal bilinear form are provided. A priori error estimate
under energy norm and optimal error estimate under $L^{2}$ norm are obtained
for DG methods of the different formulations. Finally, the performed numerical
examples confirm the optimal convergence order of the different formulations.
