In this paper, both semidiscrete 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 $L^2$ error estimates for the semidiscrete 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. Numerical examples are discussed at the end which conform our theoretical results.

翻译：本文研究了带非光滑初值的二维非定常Navier-Stokes方程在惩罚条件下的半离散和全离散有限元方法。时间离散采用一阶向后欧拉法，空间离散采用协调有限元法。在数据满足实际假定条件下，推导了速度和压力的半离散及全离散逼近的最优$L^2$误差估计。证明的主要工具是对惩罚Stokes算子逆的恰当运用、负范数估计以及时间加权估计。最后通过数值算例验证了理论结果。