Convergence rates for the numerical approximation of the 2D stochastic Navier–Stokes equations

Dominic Breit, Alan Dodgson

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Abstract

We study stochastic Navier–Stokes equations in two dimensions with respect to periodic boundary conditions. The equations are perturbed by a nonlinear multiplicative stochastic forcing with linear growth (in the velocity) driven by a cylindrical Wiener process. We establish convergence rates for a finite-element based space-time approximation with respect to convergence in probability (where the error is measured in the Lt∞Lx2∩Lt2Wx1,2-norm). Our main result provides linear convergence in space and convergence of order (almost) 1/2 in time. This improves earlier results from Carelli and Prohl (SIAM J Numer Anal 50(5):2467–2496, 2012) where the convergence rate in time is only (almost) 1/4. Our approach is based on a careful analysis of the pressure function using a stochastic pressure decomposition.

Original languageEnglish
Pages (from-to)553-578
Number of pages26
JournalNumerische Mathematik
Volume147
Issue number3
Early online date11 Feb 2021
DOIs
Publication statusPublished - Mar 2021

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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