Strong rates of convergence of space-time discretization schemes for the 2D Navier–Stokes equations with additive noise

• Published in: Stochastics and dynamics Vol. 22; no. 2
• Main Authors: Bessaih, Hakima, Millet, Annie
• Format: Journal Article
• Language: English
• Published: Singapore World Scientific Publishing Company 01.03.2022; World Scientific Publishing Co. Pte., Ltd; World Scientific Publishing
• Series: Special Issue on Modern Topics on Stochastic Dynamics, to celebrate Bjorn Schmalfuss' birthday
• Subjects: Convergence; Finite element method; Fluid flow; Mathematical analysis; Mathematics; Navier-Stokes equations; Probability; Spacetime; Toruses; Stochastic Navier–Stokes equations; exponential moments; strong convergence; finite elements; implicit time discretization; numerical schemes; Stochastic Navier-Sokes equations
• ISSN: 0219-4937; 1793-6799
• DOI: 10.1142/S0219493722400056

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 ∈ [ 0 , 1 / 2 ) in time and 1 in space. Let us mention that the coefficient η is equal to the time regularity of the solution with values in 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.