Rates of Convergence for Discretizations of the Stochastic Incompressible Navier--Stokes Equations

We show strong convergence with rates for an implicit time discretization, a semi-implicit time discretization, and a related finite element based space-time discretization of the incompressible Navier--Stokes equations with multiplicative noise in two space dimensions. We use higher moments of computed iterates to optimally bound the error on a subset Ω_κ of the sample space Ω, where corresponding paths are bounded in a proper function space, and P[Ω_κ] → 1 holds for vanishing discretization parameters. This implies convergence in probability with rates, and motivates a practicable acception/rejection criterion to overcome possible pathwise explosion behavior caused by the nonlinearity. It turns out that it is the interaction of Lagrange multipliers with the stochastic forcing in the scheme which limits the accuracy of general discretely LBB-stable space discretizations, and strategies to overcome this problem are proposed.