Error estimates for spatially discrete approximations of semilinear parabolic equations with initial data of low regularity
Michel Crouzeix, Vidar Thomée, Lars B. Wahlbin · Mathematics of Computation · 1989
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.