\(H^2\)-Stability of the First Order Fully Discrete Schemes for the Time-Dependent Navier–Stokes Equations

This paper considers the \(H^2\)-stability results for the first order fully discrete schemes based on the mixed finite element method for the time-dependent Navier–Stokes equations with the initial data \(u_0\in H^\alpha \) with \(\alpha =0,~1\) and 2. A mixed finite element method is used to the spatial discretization of the Navier–Stokes equations, and the temporal treatments of the spatial discrete Navier–Stokes equations are the first order implicit, semi-implicit, implicit/explicit(the semi-implicit/explicit in the case of \({\alpha }=0\)) and explicit schemes. The \(H^2\)-stability results of the schemes are provided, where the first order implicit and semi-implicit schemes are the \(H^2\)-unconditional stable, the first order explicit scheme is the \(H^2\)-conditional stable, and the implicit/explicit scheme (the semi-implicit/explicit scheme in the case of \({\alpha }=0\)) is the \(H^2\)-almost unconditional stable. Moreover, this paper makes some numerical investigations of the \(H^2\)-stability results for the first order fully discrete schemes for the time-dependent Navier–Stokes equations. Through a series of numerical experiments, it is verified that the numerical results are shown to support the developed \(H^2\)-stability theory.