论文标题
$ l^p $ - 解决方案估计有限元近似
$L^p$-resolvent estimate for finite element approximation of the Stokes operator
论文作者
论文摘要
在本文中，我们将展示$ l^p $ - 分辨率估算的估计值，用于$ p \ in \ in \ left（\ frac {2n} {n+2}，\ frac {2n} {2n} {n-2} {n-2} \ right）$ n \ n \ ge 2 $ domension domension的有限元近似值。可以预期，该估计值可以应用于非平稳navier方程的有限元近似值的误差估计，因为在该方向上的研究在非线性抛物线方程的数值分析中成功。为了得出分解估计，我们通过离散的外力引入了Stokes Resolvent问题的解决方案。然后，我们根据一种新的本地化技术获得局部能源误差估计，并建立全局$ l^p $ -type错误估计。 $ p $的限制是由在本地能源误差估算中出现的低阶项的处理引起的。我们的结果可能是非平稳navier的有限元方法的$ l^p $理论的突破。
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.
