Convergence rates for the numerical approximation of the 2D stochastic Navier–Stokes equations
Convergence rates for the numerical approximation of the 2D stochastic Navier–Stokes equations
复制标题
二维随机纳维-斯托克斯方程数值近似的收敛率
作者：
D. Breit;Alan Dodgson
我们研究关于周期性边界条件的二维随机纳维-斯托克斯方程。这些方程受到圆柱维纳过程驱动的线性增长（速度）的非线性乘法随机力的扰动。我们建立基于有限元的时空近似的收敛速率（其中误差在 Lt∞Lx2∩Lt2Wx1,2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} 中测量setlength{oddsidemargin}{-69pt} egin{文档}$$L^infty _tL^2_xcap L^2_tW^{1,2}_x$$end{文档}-范数）。我们的主要结果提供了空间上的线性收敛和时间上（几乎）1/2 阶的收敛。这改进了 Carelli 和 Prohl 的早期结果（SIAM J Numer Anal 50(5):2467–2496, 2012），其中时间收敛率仅为（几乎）1/4。我们的方法基于使用随机压力分解对压力函数进行仔细分析。
We study stochastic Navier–Stokes equations in two dimensions with respect to periodic boundary conditions. The equations are perturbed by a nonlinear multiplicative stochastic forcing with linear growth (in the velocity) driven by a cylindrical Wiener process. We establish convergence rates for a finite-element based space-time approximation with respect to convergence in probability (where the error is measured in the Lt∞Lx2∩Lt2Wx1,2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$L^infty _tL^2_xcap L^2_tW^{1,2}_x$$end{document}-norm). Our main result provides linear convergence in space and convergence of order (almost) 1/2 in time. This improves earlier results from Carelli and Prohl (SIAM J Numer Anal 50(5):2467–2496, 2012) where the convergence rate in time is only (almost) 1/4. Our approach is based on a careful analysis of the pressure function using a stochastic pressure decomposition.