Space-Time Approximation of Local Strong Solutions to the 3D Stochastic Navier–Stokes Equations

We consider the 3D stochastic Navier–Stokes equation on the torus. Our main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution. We prove optimal convergence rates for the energy error with respect to convergence in probability, that is convergence of order (up to) 1 in space and of order (up to) 1/2 in time. The result holds up to the possible blow-up of the (time-discrete) solution. Our approach is based on discrete stopping times for the (time-discrete) solution.