Besides the mathematical interests, stochastic Stokes and Navier-Stokes equations have been proposed to study turbulence flow under random forcing. Even in the simplest setting, their PDE solutions have very low regularity in time (and in space), which then poses a significant challenge for developing efficient and convergent numerical methods for the  stochastic Stokes and Navier-Stokes equations.  In particular,  the most natural and popular class of numerical methods for those equations, namely mixed finite element methods, had not been proven to work. In this talk I shall present some recent developments in mixed finite element methods for the Stokes and Navier-Stokes equations with multiplicative noise. I shall highlight the establishment of the continuous and discrete stochastic inf-sup conditions and the strong convergence not only for the velocity approximation but also for the pressure approximation, as well as the new analysis techniques used to obtain these results. Numerical experiments will also be presented to validate the theoretical results.

凤小兵, 美国田纳西大学数学系终身教授, 数学系副系主任, 数学系研究生学部主任; 西北工业大学长江讲座教授。1983年和1985年分别获西安交大计算数学学士和硕士学位, 1992年获美国普渡大学应用和计算数学博学位。凤小兵教授长期从事线性, 特别是非线性偏微分方程及其数值解法和算法的研究, 并取得了一系列国际领先的成果。在"SIAM Review",  "SIAM J. Numerical Analysis", "Mathematics of Computation", "Numerische Mathematik" 等国际一流专业学术期刊上发表论文100余篇。