Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations

In this article, a class of two-dimensional Riesz space fractional diffusion equations is considered. Some fractional spaces are established and some equivalences between fractional derivative spaces and fractional Sobolev space are presented. By the Galerkin finite element method and backward difference method, a fully discrete scheme is obtained. According to Lax–Milgram theorem, the existence and uniqueness of the solution to the fully discrete scheme are investigated. The stability and convergence of the scheme are also derived. Finally, some numerical examples are given for verification of our theoretical analysis.