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

Abstract

This paper considers the $$H^2$$ 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 $$u_0\in H^\alpha $$ u 0 ? H ? with $$\alpha =0,~1$$ ? = 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 $${\alpha }=0$$ ? = 0 ) and explicit schemes. The $$H^2$$ H 2 -stability results of the schemes are provided, where the first order implicit and semi-implicit schemes are the $$H^2$$ H 2 -unconditional stable, the first order explicit scheme is the $$H^2$$ H 2 -conditional stable, and the implicit/explicit scheme (the semi-implicit/explicit scheme in the case of $${\alpha }=0$$ ? = 0 ) is the $$H^2$$ H 2 -almost unconditional stable. Moreover, this paper makes some numerical investigations of the $$H^2$$ 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 $$H^2$$ H 2 -stability theory.