  The subdiffusion equation with a Caputo fractional derivative of order
$\alpha\in(0,1)$ in time arises in a wide variety of practical applications,
and it is often adopted to model anomalous subdiffusion processes in
heterogeneous media. The L1 scheme is one of the most popular and successful
numerical methods for discretizing the Caputo fractional derivative in time.
The scheme was analyzed earlier independently by Lin and Xu (2007) and Sun and
Wu (2006), and an $O(\tau^{2-\alpha})$ convergence rate was established, under
the assumption that the solution is twice continuously differentiable in time.
However, in view of the smoothing property of the subdiffusion equation, this
regularity condition is restrictive, since it does not hold even for the
homogeneous problem with a smooth initial data. In this work, we revisit the
error analysis of the scheme, and establish an $O(\tau)$ convergence rate for
both smooth and nonsmooth initial data. The analysis is valid for more general
sectorial operators. In particular, the L1 scheme is applied to one-dimensional
space-time fractional diffusion equations, which involves also a
Riemann-Liouville derivative of order $\beta\in(3/2,2)$ in space, and error
estimates are provided for the fully discrete scheme. Numerical experiments are
provided to verify the sharpness of the error estimates, and robustness of the
scheme with respect to data regularity.
