Optimally convergent mixed finite element methods for the stochastic Stokes equations

Abstract We propose some new mixed finite element methods for the time-dependent stochastic Stokes equations with multiplicative noise, which use Helmholtz decomposition of driving noise. It is known (Langa, J. A., Real, & Simon, (2003) Existence and regularity pressure Navier--Stokes equations. Appl. Math. Optim., 48, 195--210) that solution has low regularity, manifests in suboptimal convergence rates well-known inf-sup stable numerical simulations; see Feng X., Qiu, H. (Analysis fully discrete arXiv:1905.03289v2 [math.NA]). show eliminating this gradient part from noise scheme leads to optimally convergent conceptual idea may be used retool are well deterministic setting, including stabilization methods, so their optimal properties can still maintained setting. Computational experiments also provided validate theoretical results illustrate usefulness proposed approach.