We study finite element based space-time discretisations of the incompressible Navier-Stokes equations with noise. In three dimensions, sequences of numerical solutions construct weak martingale solutions for vanishing discretisation parameters. In the two dimensional case, numerical solutions converge to the unique strong solution.
For the stochastic incompressible time-dependent Stokes equation, we study different time-splitting methods that decouple the computation of velocity and pressure iterates in every iteration step. Optimal strong convergence is shown for Chorin's time-splitting scheme in the case of solenoidal noise, while computational counterexamples show a poor convergence behavior in the case of general stochastic forcing. This sub-optimal performance may be traced back to the non-regular pressure process in the case of general noise. A modified version of the deterministic time-splitting method that distinguishes between the deterministic and stochastic pressure removes this deficiency, leading to optimal convergence behavior.
We study equations to describe incompressible generalized Newtonian fluids, where the extra stress tensor satisfies a nonstandard anisotropic asymptotic growth condition. An implicit finite element discretization, and a simple, fully practical fixed-point scheme with proper thresholding criterion are proposed, and convergence towards weak solutions of the limiting problem is shown. Computational experiments are included, which motivate nontrivial fluid flow behavior.
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