ABSTRACT

We consider the discretization of a class of nonlinear parabolic equations in a usual triple of Hilbert spaces $V\subset H=H'\subset V'$ by backward difference formula (BDF) methods. Local stability of general multistep methods in the discrete $L^\infty (H)$ and $L^2(V)$ norms was recently established under sharp stability conditions. In the case of BDF methods the stability estimate can be improved in the sense that only the $H$-norm, but not the $V$-norm, of the starting approximations enters in the stability bound. The improved stability leads to optimal order error estimates under relaxed accuracy requirements on the starting approximations.