论文标题
最佳分析随机Stokes方程的有限元方法
Optimal analysis of finite element methods for the stochastic Stokes equations
论文作者
论文摘要
随机Stokes方程的数值分析仍然具有挑战性，即使对相应的确定性方程进行了很好的做法。特别是，随机stokes方程的有限元方法的预先存在的误差估计{在$ l^\ infty（0，t; l^2（ω; l^2））$ norm} $} $} $} $} $} $} $} $} $} $}均与空间离散化有关的订单减少。这些完全离散的方案获得的最佳收敛结果仅是时间上的半顺序，而在太空中的一阶则是一阶，这在传统意义上并不是最佳的空间。本文的目的是在$ l^\ infty（0，t; t; l^2（ω; l^2））中建立$ O（τ^{1/2}+ h^2）$的强大收敛性，以近似速度，并且强烈的融合$ o（the 1/2}+ h）$ o（ t; l^2（ω; l^2））$ norm近似于压力的时间积分，其中$τ$和$ h $分别表示时间步长大小和空间网格大小。误差估计值是本文中考虑的空间离散化的最佳顺序（带有迷你元素），并且与数值实验一致。该分析基于完全离散的STOKES半群技术和相应的新估计值。
Numerical analysis for the stochastic Stokes equations is still challenging even though it has been well done for the corresponding deterministic equations. In particular, the pre-existing error estimates of finite element methods for the stochastic Stokes equations { in the $L^\infty(0, T; L^2(Ω; L^2))$ norm} all suffer from the order reduction with respect to the spatial discretizations. The best convergence result obtained for these fully discrete schemes is only half-order in time and first-order in space, which is not optimal in space in the traditional sense. The objective of this article is to establish strong convergence of $O(τ^{1/2}+ h^2)$ in the $L^\infty(0, T; L^2(Ω; L^2))$ norm for approximating the velocity, and strong convergence of $O(τ^{1/2}+ h)$ in the $L^{\infty}(0, T;L^2(Ω;L^2))$ norm for approximating the time integral of pressure, where $τ$ and $h$ denote the temporal step size and spatial mesh size, respectively. The error estimates are of optimal order for the spatial discretization considered in this article (with MINI element), and consistent with the numerical experiments. The analysis is based on the fully discrete Stokes semigroup technique and the corresponding new estimates.
