Numerical analysis of a semilinear fractional diffusion equation

This paper considers the numerical analysis of a semilinear fractional diffusion equation with nonsmooth initial data. A new Gr\"onwall's inequality and its discrete version are proposed. By the two inequalities, error estimates in three Sobolev norms are derived for a spatial semi-discretization and a full discretization, which are optimal with respect to the regularity of the solution. A sharp temporal error estimate on graded temporal grids is also rigorously established. In addition, the spatial accuracy $\scriptstyle O(h^2(t^{-\alpha} + \ln(1/h)\!)\!)$ in the pointwise $\scriptstyle L^2(\Omega)$-norm is obtained for a spatial semi-discretization. Finally, several numerical results are provided to verify the theoretical results.