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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 in the pointwise -norm is obtained for a spatial semi-discretization. Finally,
several numerical results are provided to verify the theoretical results