Numerical analysis of fully discrete finite element methods for the stochastic Navier-Stokes equations with multiplicative noise

摘要

Previous work on the stability and convergence analysis of the finite element methods for the deterministic Navier-Stokes equations was carried out under the uniqueness condition. In this paper, the corresponding results of fully discrete finite element method are developed for the stochastic Navier-Stokes equations with multiplicative noise. The stability and error estimates for velocity in expectation are rigorously proved, and the L~2-norm in a time-averaged version for pressure is provided with convergence rates in spatial and temporal form O((△t+h)/(△t)~1/2). Finally, numerical experiments are given to illustrate the features of the proposed numerical method and verify the theoretical claims.