Space-time Euler discretization schemes for the stochastic 2D Navier–Stokes equations

• Published in: Stochastic partial differential equations : analysis and computations Vol. 10; no. 4; pp. 1515 - 1558
• Main Authors: Bessaih, Hakima, Millet, Annie
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2022; Springer Nature B.V
• Subjects: Approximation; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Convergence; Diffusion coefficient; Diffusion rate; Discretization; Fluid flow; Mathematics; Mathematics and Statistics; Navier-Stokes equations; Numerical Analysis; Partial Differential Equations; Perturbation; Polynomials; Probability; Probability Theory and Stochastic Processes; Statistical Theory and Methods; Toruses; Exponential moments; 76M35; Finite elements; Strong convergence; Secondary 76D06; Primary 60H15; Implicit time discretization; Stochastic Navier–Stokes equations; Euler schemes; 60H35; Stochastic Navier-Sokes equations; exponential moments; strong convergence; finite elements; implicit time discretization
• ISSN: 2194-0401; 2194-041X
• DOI: 10.1007/s40072-021-00217-7

We prove that the implicit time Euler scheme coupled with finite elements space discretization for the 2D Navier–Stokes equations on the torus subject to a random perturbation converges in L 2 ( Ω ) , and describe the rate of convergence for an H 1 -valued initial condition. This refines previous results which only established the convergence in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic Navier–Stokes equations and convergence of a localized scheme, we can prove strong convergence of this space-time approximation. The speed of the L 2 ( Ω ) -convergence depends on the diffusion coefficient and on the viscosity parameter. In case of Scott–Vogelius mixed elements and for an additive noise, the convergence is polynomial.