Unconditionally optimal error estimates of BDF2 Galerkin method for semilinear parabolic equation

In this paper, a 2-step backward differentiation formula (BDF2) Galerkin method is investigated for semilinear parabolic equation. More precisely, the second-order time-stepping scheme is used for time discretization and the piecewise linear continuous Galerkin method is employed for spatial discretization, respectively. Optimal error estimates in L-2 and H-1-norms are obtained without any restriction on the time-step size, while previous works always require certain conditions on time step-size. The key to our analysis is to derive a uniform boundness of the numerical solution in energy norm so as to avoid the inverse inequality used in the usual convergence analysis of the finite element methods. Numerical experiments are carried out to confirm the theoretical analysis.