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articleFree Access
Strong rates of convergence of space-time discretization schemes for the 2D Navier–Stokes equations with additive noise
- Hakima Bessaih and
- Annie Millet
Stochastics and Dynamics26 Jan 2022
Preview Abstract
We consider the strong solution of the 2D Navier–Stokes equations in a torus subject to an additive noise. We implement a fully implicit time numerical scheme and a finite element method in space. We prove that the space-time rate of convergence is the “optimal” one, namely, $η \in [0, 1 / 2)$ $η \in [0, 1 / 2)$ in time and 1 in space. Let us mention that the coefficient $η$ $η$ is equal to the time regularity of the solution with values in $L^{2}$ $𝕃^{2}$ . Our method relies on the existence of finite exponential moments for both the solution and its time approximation. Unlike previous results, our main new idea is the use of a discrete Grönwall lemma for the error estimate without any localization.

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