Error estimates for spatially discrete approximations of semilinear parabolic equations with initial data of low regularity

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