A posteriori error estimates of finite element method for the time-dependent Navier-Stokes equations

In this paper, we consider the posteriori error estimates of Galerkin finite element method for the unsteady Navier-Stokes equations. By constructing the approximate Navier-Stokes reconstructions, the errors of velocity and pressure are split into two parts. For the estimates of time part, the energy method and other skills are used, for the estimates of spatial part, the well-developed theoretical analysis of posteriori error estimates for the elliptic problem can be adopted. More important, the error estimates of time part can be controlled by the estimates of spatial part. As a consequence, the posteriori error estimates in L∞(0, T; L2(Ω)), L∞(0, T; H1(Ω)) and L2(0, T; L2(Ω)) norms for velocity and pressure are derived in both spatial discrete and time-space fully discrete schemes.