Rates of Convergence for Discretizations of the Stochastic Incompressible Navier-Stokes Equations
Rates of Convergence for Discretizations of the Stochastic Incompressible Navier-Stokes Equations
复制标题
随机不可压缩纳维-斯托克斯方程离散化的收敛率
DOI：
10.1137/110845008
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
A. Prohl
中科院分区：
文献类型：
--
作者：
Erich Carelli;A. Prohl
我们展示了隐式时间离散化、半隐式时间离散化以及二维空间中具有乘性噪声的不可压缩Navier—Stokes方程的相关基于有限元的时空离散化的强收敛率。我们使用较高的计算迭代矩来最优地约束样本空间$\Omega$的子集$\Omega_\kappa$上的误差，其中对应的路径在适当的函数空间中有界，并且$\mathbb{P}[\Omega_\kappa] \to 1$用于消失的离散化参数。这意味着概率随速率的收敛，并激发了一个可行的接受/拒绝准则，以克服可能由非线性引起的路径爆炸行为。结果表明，是拉格朗日乘子与随机强迫的相互作用限制了一般离散lbb稳定空间离散化的精度，并提出了克服这一问题的策略。
We show strong convergence with rates for an implicit time discretization, a semi-implicit time discretization, and a related finite element based space-time discretization of the incompressible Navier--Stokes equations with multiplicative noise in two space dimensions. We use higher moments of computed iterates to optimally bound the error on a subset $\Omega_\kappa$ of the sample space $\Omega$, where corresponding paths are bounded in a proper function space, and $\mathbb{P}[\Omega_\kappa] \to 1$ holds for vanishing discretization parameters. This implies convergence in probability with rates, and motivates a practicable acception/rejection criterion to overcome possible pathwise explosion behavior caused by the nonlinearity. It turns out that it is the interaction of Lagrange multipliers with the stochastic forcing in the scheme which limits the accuracy of general discretely LBB-stable space discretizations, and strategies to overcome this problem are proposed.