Semidiscrete finite element methods for a semilinear parabolic equation in
R
d
{R^d}
,
d
≤
3
d \leq 3
, were considered by Johnson, Larsson, Thomée, and Wahlbin. With h the discretization parameter, it was proved that, for compatible and bounded initial data in
H
α
{H^\alpha }
, the convergence rate is essentially
O
(
h
2
+
α
)
O({h^{2 + \alpha }})
for t positive, and for
α
=
0
\alpha = 0
this was seen to be best possible. Here we shall show that for
0
≤
α
>
2
0 \leq \alpha > 2
the convergence rate is, in fact, essentially
O
(
h
2
+
2
α
)
O({h^{2 + 2\alpha }})
, which is sharp.