Optimally Convergent Mixed Finite Element Methods for the Stochastic Stokes Equations

We propose some new mixed finite element methods for the time dependent stochastic Stokes equations with multiplicative noise, which use the Helmholtz decomposition of the driving multiplicative noise. It is known [16] that the pressure solution has a low regularity, which manifests in sub-optimal convergence rates for well-known inf-sup stable mixed finite element methods in numerical simulations, see [10]. We show that eliminating this gradient part from the noise in the numerical scheme leads to optimally convergent mixed finite element methods, and that this conceptual idea may be used to retool numerical methods that are well-known in the deterministic setting, including pressure stabilization methods, so that their optimal convergence properties can still be maintained in the stochastic setting. Computational experiments are also provided to validate the theoretical results and to illustrate the conceptional usefulness of the proposed numerical approach.