二维随机纳维-斯托克斯方程的数值分析：正则性与空间半离散化
Numerical Analysis of 2D Stochastic Navier--Stokes Equations with Transport Noise: Regularity and Spatial Semidiscretization
AI总结：
本文研究了带有传输噪声的二维随机纳维-斯托克斯方程的数值分析，通过引入新的平滑算子，解决了低正则性问题，得出了空间半离散化的均方收敛估计。
摘要：
本文建立了带有传输噪声和无滑边界条件的二维随机纳维-斯托克斯系统的空间有限元离散化的强收敛率。主要挑战来自于解在空间上缺乏$D(A)$-正则性（其中$A$是斯托克斯算子），这阻止了标准误差分析技术的应用。在小噪声假设下，我们证明弱解满足$u \in L^2(Ω; C([0,T]; \dot{H}_σ^{\varrho}) \cap L^2(0,T; \dot{H}_σ^{1+\varrho}))$对于某些$\varrho \in (0,\tfrac{1}{2})$。为了解决数值分析中的低正则性问题，我们引入了一个新的平滑算子$J_{h,α} = A_h^α\mathcal{P}_h A^{-α}$，其中$A_h$是离散斯托克斯算子，$\mathcal{P}_h$是离散霍尔兹投影。这个工具使得对MINI元空间半离散化的完整误差分析成为可能，得出均方收敛估计$ \|u - u_h\|_{L^2(Ω; C([0,T]; L^2(\mathcal O;\mathbb{R}^2)))} + \|\nabla(u - u_h)\|_{L^2(Ω\times (0,T); L^2(\mathcal{O};\mathbb{R}^{2\times2}))} \leqslant c\, h^{\varrho} \log\big(1 + \frac{1}{h}\big) $。该框架可以扩展到更广泛的随机流体模型，具有粗糙噪声和狄利克雷边界条件。
英文摘要：
This paper establishes strong convergence rates for the spatial finite element discretization of a two-dimensional stochastic Navier--Stokes system with transport noise and no-slip boundary conditions on a convex polygonal domain. The main challenge arises from the lack of spatial \(D(A)\)-regularity of the solution (where \(A\) is the Stokes operator), which prevents the application of standard error analysis techniques. Under a small-noise assumption, we prove that the weak solution satisfies \[ u \in L^2\bigl(Ω; C([0,T]; \dot{H}_σ^{\varrho}) \cap L^2(0,T; \dot{H}_σ^{1+\varrho})\bigr) \] for some \(\varrho \in (0,\tfrac{1}{2})\). To address the low regularity in the numerical analysis, we introduce a novel smoothing operator \(J_{h,α} = A_h^α\mathcal{P}_h A^{-α}\) with \(α\in (0,1)\), where \(A_h\) is the discrete Stokes operator and \(\mathcal{P}_h\) the discrete Helmholtz projection. This tool enables a complete error analysis for a MINI-element spatial semidiscretization, yielding the mean-square convergence estimate \[ \|u - u_h\|_{L^2(Ω; C([0,T]; L^2(\mathcal O;\mathbb{R}^2)))} + \|\nabla(u - u_h)\|_{L^2(Ω\times (0,T); L^2(\mathcal{O};\mathbb{R}^{2\times2}))} \leqslant c\, h^{\varrho} \log\big(1 + \frac{1}{h}\big). \] The framework can be extended to broader stochastic fluid models with rough noise and Dirichlet boundary conditions.
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