Abstract
A reaction-diffusion problem with a Caputo time derivative of order \(\alpha \in (0,1)\) is considered. The solution of such a problem has in general a weak singularity near the initial time \(t = 0\). Some new pointwise bounds on certain derivatives of this solution are derived. The numerical method of the paper uses the well-known L1 discretisation in time on a graded mesh and a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh. Discrete stability of the computed solution is proved. The error analysis is based on a non-trivial projection into the finite element space, which for the first time extends the analysis of the DDG method to non-periodic boundary conditions. The final convergence result implies how an optimal grading of the temporal mesh should be chosen. Numerical results show that our analysis is sharp.
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Communicated by Jan Hesthaven.
The research of Martin Stynes is supported in part by the National Natural Science Foundation of China under Grants 91430216 and NSAF-U1530401.
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Huang, C., Stynes, M. & An, N. Optimal \(L^\infty (L^2)\) error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem. Bit Numer Math 58, 661–690 (2018). https://doi.org/10.1007/s10543-018-0707-z
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DOI: https://doi.org/10.1007/s10543-018-0707-z