带乘性噪声随机Navier-斯托克斯方程的全离散混合有限元格式分析
Analysis of fully discrete mixed finite element scheme for Stochastic Navier-Stokes equations with multiplicative noise
摘要：
本文研究二维带乘性噪声的随机不可压Navier-斯托克斯方程在周期边界条件下的解。基于乘性噪声的Helmholtz分解，提出了半离散和全离散时间步进算法。得到了基于时空近似的混合有限元法关于速度和压力的依概率收敛的收敛速度。通过建立稳定性和利用负范数技巧，证明了速度误差的$H^1$和$L^2$范数的部分期望最优收敛.
英文摘要：
This paper is concerned with stochastic incompressible Navier-Stokes equations with multiplicative noise in two dimensions with respect to periodic boundary conditions. Based on the Helmholtz decomposition of the multiplicative noise, semi-discrete and fully discrete time-stepping algorithms are proposed. The convergence rates for mixed finite element methods based time-space approximation with respect to convergence in probability for the velocity and the pressure are obtained. Furthermore, with establishing some stability and using the negative norm technique, the partial expectations of the $H^1$ and $L^2$ norms of the velocity error are proved to converge optimally.
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