Optimal Error Estimates of the Penalty Finite Element Method for the Unsteady Navier–Stokes Equations with Nonsmooth Initial Data

In this paper, both semi-discrete and fully discrete finite element methods are analyzed for the penalized two-dimensional unsteady Navier–Stokes equations with nonsmooth initial data. First order backward Euler method is applied for the time discretization, whereas conforming finite element method is used for the spatial discretization. Optimal \(L^2\) $L^2$ error estimates for the semi-discrete as well as the fully discrete approximations of the velocity and of the pressure are derived for realistically assumed conditions on the data. The main ingredient in the proof is the appropriate exploitation of the inverse of the penalized Stokes operator, negative norm estimates and time weighted estimates. Two numerical examples one in 2D and one in 3D are presented whose results are conforming our theoretical findings. Finally, computational experiments on benchmark problem: one on lid driven cavity problem and other on flow around a cylinder with low viscosity are discussed.