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arXiv · 1909.00016

Numerical analysis of a semilinear fractional diffusion equation

Binjie Li·Tao Wang·Xiaoping Xie
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Abstract

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.

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Binjie Li, Tao Wang, Xiaoping Xie. 2019-08-30. Numerical analysis of a semilinear fractional diffusion equation. https://arxiv.org/abs/1909.00016

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