Blow-up of error estimates in time-fractional initial-boundary value problems

Time-fractional initial-boundary value problems of the form $D_t^\alpha u-p \varDelta u +cu=f$ are considered, where $D_t^\alpha u$ is a Caputo fractional derivative of order $\alpha \in (0,1)$ and the spatial domain lies in $\mathbb{R}^d$ for some $d\in \{1,2,3\}$. As $\alpha \to 1^-$ we prove that the solution $u$ converges, uniformly on the space-time domain, to the solution of the classical parabolic initial-boundary value problem where $D_t^\alpha u$ is replaced by $\partial u/\partial t$. Nevertheless, most of the rigorous analyses of numerical methods for this time-fractional problem have error bounds that blow up as $\alpha \to 1^-$, as we demonstrate. We show that in some cases these analyses can be modified to obtain robust error bounds that do not blow up as $\alpha \to 1^-$.