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。由于该方向的研究在非线性抛物型方程的数值分析中已取得成功，预期该估计可用于非定常Navier--Stokes方程有限元逼近的误差估计。为推导预解估计，我们引入带离散外力的Stokes预解问题的解。随后，根据一种新颖的局部化技术得到局部能量误差估计，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。我们的结果可能成为非定常Navier--Stokes方程有限元方法$L^p$理论的一个突破。