  We investigate semi-discrete numerical schemes based on the standard Galerkin
and lumped mass Galerkin finite element methods for an initial-boundary value
problem for homogeneous fractional diffusion problems with non-smooth initial
data. We assume that $\Omega\subset \mathbb{R}^d$, $d=1,2,3$ is a convex
polygonal (polyhedral) domain. We theoretically justify optimal order error
estimates in $L_2$- and $H^1$-norms for initial data in $H^{-s}(\Omega),~0\le s
\le 1$. We confirm our theoretical findings with a number of numerical tests
that include initial data $v$ being a Dirac $\delta$-function supported on a
$(d-1)$-dimensional manifold.
