  A time-stepping L1 scheme for subdiffusion equation with a Riemann--Liouville
time-fractional derivative is developed and analyzed. This is the first paper
to show that the L1 scheme for the model problem under consideration is
second-order accurate (sharp error estimate) over nonuniform time-steps. The
established convergence analysis is novel, innovative and concise. For
completeness, the L1 scheme is combined with the standard Galerkin finite
elements for the spatial discretization, which will then define a
fully-discrete numerical scheme. The error analysis for this scheme is also
investigated. To support our theoretical contributions, some numerical tests
are provided at the end. The considered (typical) numerical example suggests
that the imposed time-graded meshes assumption can be further relaxed.
