Viscosity explicit analysis for finite element methods of time-dependent Navier–Stokes equations

In this paper, the viscosity explicit stability and error estimate for the finite element method of the time-dependent Navier–Stokes equations are deduced. Firstly, the viscosity explicit analysis is derived for a linearized auxiliary problem. Then, by introducing an iterative procedure and investigating its convergence, the results are extended to the time-dependent Navier–Stokes equations. The new error estimate for the finite element method is also derived. The analysis shows that, under some assumptions on the given data, both the stability and the error for the finite element approximation of the time-dependent Navier–Stokes equations depend on the viscosity coefficient with algebraic orders, which is different from the exponential dependence in the literature. Moreover, the relationship between the body force’s regularity and the stability is also analyzed in detail.