  We consider the initial boundary value problem for the homogeneous
time-fractional diffusion equation $\partial^\alpha_t u - \De u =0$ ($0< \alpha
< 1$) with initial condition $u(x,0)=v(x)$ and a homogeneous Dirichlet boundary
condition in a bounded polygonal domain $\Omega$. We shall study two
semidiscrete approximation schemes, i.e., Galerkin FEM and lumped mass Galerkin
FEM, by using piecewise linear functions. We establish optimal with respect to
the regularity of the solution error estimates, including the case of nonsmooth
initial data, i.e., $v \in L_2(\Omega)$.
