Backward difference formulae and spectral Galerkin methods for the Riesz space fractional diffusion equation

Approximating Riesz space fractional diffusion equation in time by k-step backward difference formula and in space by spectral Galerkin method, we establish a fully discrete scheme with high order both in time and in space. For k≤5, we prove the stability of full discretization and obtain the error estimate with order O(τ^k+N ^{[Formula presented] −m}), which depends only on the regularity of initial value and right-hand function. Moreover, we extend the proposed method to two dimensional case and derive similar results. Finally, we illustrate the theoretical estimates by numerical examples.