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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2025, Vol. 42 ›› Issue (4): 632-644.doi: 10.3969/j.issn.1005-3085.2025.04.003cstr: 32411.14.cjem.CN61-1269/O1.2025.04.003

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随机Navier-Stokes方程的稳定化有限元方法研究

何佳琴,  贾宏恩   

  1. 太原理工大学数学学院，太原  030024
  • 收稿日期:2022-12-21 接受日期:2023-04-27 出版日期:2025-08-15 发布日期:2025-10-15
  • 通讯作者: 贾宏恩 E-mail: jiahongen@aliyun.com
  • 基金资助:
    山西省学术委员会基金 (2021029)；2021 山西省科技合作与交流专项基金 (202104041101019).

A Study of the Stabilized Finite Element Method for the Stochastic Navier-Stokes Equations

HE Jiaqin,  JIA Hongen$   

  1. College of Mathematics, Taiyuan University of Technology, Taiyuan 030024
  • Received:2022-12-21 Accepted:2023-04-27 Online:2025-08-15 Published:2025-10-15
  • Contact: H. Jia. E-mail address: jiahongen@aliyun.com
  • Supported by:
    The Shanxi Scholarship Council of China (2021029); the 2021 Shanxi Science and Technology Cooperation and Exchange Special Program (202104041101019).

摘要：

针对带乘性噪音的随机 Navier-Stokes 方程的压力稳定方法进行了研究，对于速度压力空间采用一阶等阶有限元进行逼近。该方法通过引入稳定化参数，解耦速度压力变量，解决了有限元空间对选取的问题。从理论上证明了压力稳定方法的稳定性和收敛性，并最终证明当稳定化参数满足一定条件时，空间离散的收敛阶可以达到最优。

关键词: 乘性噪音, 稳定化有限元方法, 随机 Navier-Stokes 方程, P1-P1 元, 全离散

Abstract:

In this paper, the pressure stabilization method of the stochastic Navier-Stokes equation with multiplicative noise is studied, and the velocity pressure space is approximated by first-order finite elements. This method solves the problem of finite element space pair selection by introducing stabilization parameters and decoupling velocity pressure variables. The stability and convergence of the pressure stabilization method are proved theoretically. Finally, it is proved that the convergence order of space discreteness can reach the optimal when the stabilization parameters meet certain conditions.

Key words: multiplicative noise, stablized finite element method, stochastic Navier-Stokes equations, P1-P1 element, fully discretization

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