In this paper, we will show the $L^p$-resolvent estimate for the finite element approximation of the Stokes operator for $p \in \left( \frac{2N}{N+2}, \frac{2N}{N-2} \right)$, where $N \ge 2$ is the dimension of the domain. It is expected that this estimate can be applied to error estimates for finite element approximation of the non-stationary Navier--Stokes equations, since studies in this direction are successful in numerical analysis of nonlinear parabolic equations. To derive the resolvent estimate, we introduce the solution of the Stokes resolvent problem with a discrete external force. We then obtain local energy error estimate according to a novel localization technique and establish global $L^p$-type error estimates. The restriction for $p$ is caused by the treatment of lower-order terms appearing in the local energy error estimate. Our result may be a breakthrough in the $L^p$-theory of finite element methods for the non-stationary Navier--Stokes equations.

翻译：本文将对Stokes算子的有限元逼近建立$L^p$预解估计，其中$p \in \left( \frac{2N}{N+2}, \frac{2N}{N-2} \right)$，$N \ge 2$为区域维数。该估计有望应用于非定常Navier-Stokes方程有限元逼近的误差估计，因为在此方向上的研究已在非线性抛物型方程的数值分析中获得成功。为推导预解估计，我们引入带有离散外力的Stokes预解问题的解。进而根据一种新型局部化技巧获得局部能量误差估计，并建立全局$L^p$型误差估计。$p$的限制源于局部能量误差估计中低阶项的处理。本文结果或将成为非定常Navier-Stokes方程有限元方法$L^p$理论的突破。