随机Navier-Stokes方程的两重网格有限元方法研究

Study of the two-level finite element method for stochastic Navier-Stokes equations

针对伴随乘性噪声的随机Navier-Stokes方程,本文提出了基于时间变量Euler-Maruyama离散格式和空间变量Taylor-Hood混合有限元格式的Oseen两重网格有限元方法。即首先在粗网格有限元空间X^(H)上处理非线性随机Navier-Stokes问题,之后在细网格有限元空间X^(h)(h<<H)上求解线性随机Stokes问题。当粗网格和细网格满足H=O(√τ^(1/4)-τ^(3/4))h的条件时,此方法与传统有限元方法相比保持几乎相同的精度,还可以节省大量的计算时间,在一定程度上提高了随机问题的求解效率。最后,通过数值实验验证了理论分析的正确性。

For the stochastic Navier-Stokes equations with multiplicative noise,the Oseen two-level finite element method is proposed based on the Euler-Maruyama discrete scheme with temporal variables and the Taylor-Hood mixed finite element scheme with space variables.Firstly,the nonlinear stochastic Navier-Stokes problem is solved in the coarse mesh finite element space X^(H),with the linear stochastic Stokes problem solved in the fine mesh finite element space X^(h)(h<<H).When the size of the coarse and fine mesh satisfies H=O(√τ^(1/4)-τ^(3/4))h,this method can not only maintain almost the same accuracy,but also save a lot of computing time;it improves the efficiency of solving stochastic problems to a certain extent.Finally,the correctness of the theoretical analysis is verified by numerical experiments.