Numerical analysis of a semilinear fractional diffusion equation

2019/08/30 by Li, Binjie, Wang, Tao, Xie, Xiaoping
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1909.00016

Abstract

This paper considers the numerical analysis of a semilinear fractional diffusion equation with nonsmooth initial data. A new Grönwall'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(h²(t^-α + ln(1/h) ) ) in the pointwise \scriptstyle L²(Ω) -norm is obtained for a spatial semi-discretization. Finally, several numerical results are provided to verify the theoretical results.

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