Optimal $$L^\infty (L^2)$$ L ∞ ( L 2 ) error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem

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.