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We consider a time discretization of incompressible Navier-Stokes equationswith spatial periodic boundary conditions in the vorticity-velocityformulation. The approximation is based on freezing the velocity on timesubintervals resulting in linear parabolic equations for vorticity.Probabilistic representations for solutions of these linear equations aregiven. At each time step, the velocity is expressed via vorticity using aformula corresponding to the Biot--Savart-type law. We show that theapproximation is divergent free and of first order. The results are extended totwo-dimensional stochastic Navier-Stokes equations with additive noise, where,in particular, we prove the first mean-square convergence order of thevorticity approximation.

Approximation of deterministic and stochastic Navier-Stokes equations in vorticity-velocity formulation