  In this paper, we consider the numerical approximation of time-fractional
parabolic problems involving Caputo derivatives in time of order $\alpha$, $0<
\alpha<1$. We derive optimal error estimates for semidiscrete Galerkin FE type
approximations for problems with smooth and nonsmooth initial data. Our
analysis relies on energy arguments and exploits the properties of the inverse
of the associated elliptic operator. We present the analysis in a general
setting so that it is easily applicable to various spatial approximations such
as conforming and nonconforming FEMs, and FEM on nonconvex domains. The finite
element approximation in mixed form is also presented and new error estimates
are established for smooth and nonsmooth initial data. Finally, an extension of
our analysis to a multi-term time-fractional model is discussed.
