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

@article{Breit2021ConvergenceRF, title={Convergence rates for the numerical approximation of the 2D stochastic Navier-Stokes equations}, author={Dominic Breit and Alan Dodgson}, journal={Numerische Mathematik}, year={2021}, volume={147}, pages={553-578} }

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 measure in the $L^\infty_tL^2_x\cap L^2_tW^{1,2}_x$-norm). Our main result provides linear…

## 9 Citations

Numerical and convergence analysis of the stochastic Lagrangian averaged Navier-Stokes equations

- Mathematics, Computer ScienceArXiv
- 2021

A finite element based space-time discretization for solving the stochastic Lagrangian averaged NavierStokes equations of incompressible fluid turbulence with multiplicative random forcing under nonperiodic boundary conditions within a bounded polygonal domain.

Error analysis for 2D stochastic Navier-Stokes equations in bounded domains

- MathematicsArXiv
- 2021

It is proved optimal convergence rates in the energy norm with respect to convergence in probability, that is convergence of order (almost) 1/2 in time and 1 in space.

Analysis of Chorin-Type Projection Methods for the Stochastic Stokes Equations with General Multiplicative Noises

- Computer ScienceArXiv
- 2020

It is proved that all spatial error constants contain a growth factor, which explains the deteriorating performance of the standard Chorin scheme when $k\to 0$ and the space mesh size is fixed as observed earlier in the numerical tests of [9].

Analysis of Fully Discrete Mixed Finite Element Methods for Time-dependent Stochastic Stokes Equations with Multiplicative Noise

- MathematicsJ. Sci. Comput.
- 2021

Strong convergence with rates is established not only for the velocity approximation but also for the pressure approximation (in a time-averaged fashion) for the time-dependent stochastic Stokes equations with multiplicative noise.

High moment and pathwise error estimates for fully discrete mixed finite element approximations of the Stochastic Stokes Equations with Multiplicative Noises

- MathematicsArXiv
- 2021

The main idea for deriving the high moment error estimates for the velocity approximation is to use a bootstrap technique starting from the second moment error estimate, which results in a pathwise error estimate which is sub-optimal in the energy norm.

Numerical analysis of 2D Navier-Stokes equations with additive stochastic forcing

- Mathematics, Computer ScienceArXiv
- 2021

This work proposes and studies a temporal, and spatio-temporal discretisation of the 2D stochastic Navier–Stokes equations in bounded domains supplemented with no-slip boundary conditions and shows strong rate (up to) 1 in probability for a correspondingDiscretisation in space and time.

Estimate Analysis of a Fully Discrete Mixed Finite Element Scheme for Stochastic Incompressible Navier-Stokes Equations with Multiplicative Noise

- MathematicsArXiv
- 2022

Abstract. This paper is concerned with stochastic incompressible Navier-Stokes equations with multiplicative noise in two dimensions with respect to periodic boundary conditions. Based on the…

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

- MathematicsStochastics and Partial Differential Equations: Analysis and Computations
- 2021

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…

Optimally Convergent Mixed Finite Element Methods for the Stochastic Stokes Equations

- MathematicsArXiv
- 2020

It is shown that eliminating this gradient part from the noise in the numerical scheme leads to optimally convergent mixed finite element methods, and that this conceptual idea may be used to retool numerical methods that are well-known in the deterministic setting, including pressure stabilization methods, so that their optimal convergence properties can still be maintained in the stochastic setting.

## References

SHOWING 1-10 OF 37 REFERENCES

Splitting up method for the 2D stochastic Navier–Stokes equations

- Mathematics
- 2013

In this paper, we deal with the convergence of an iterative scheme for the 2-D stochastic Navier–Stokes equations on the torus suggested by the Lie–Trotter product formulas for stochastic…

Rates of Convergence for Discretizations of the Stochastic Incompressible Navier-Stokes Equations

- Computer Science, MathematicsSIAM J. Numer. Anal.
- 2012

It turns out that it is the interaction of Lagrange multipliers with the stochastic forcing in the scheme which limits the accuracy of general discretely LBB-stable space discretizations, and strategies to overcome this problem are proposed.

Strong $L^2$ convergence of time numerical schemes for the stochastic two-dimensional Navier–Stokes equations

- Mathematics, Computer ScienceIMA Journal of Numerical Analysis
- 2018

We prove that some time discretization schemes for the two-dimensional Navier–Stokes equations on the torus subject to a random perturbation converge in $L^2(\varOmega )$. This refines previous…

Finite element approximation of the nonstationary Navier-Stokes problem. I : Regularity of solutions and second-order error estimates for spatial discretization

- Computer Science
- 1982

Second-order error estimates are proven for spatial discretization, using conforming or nonconforming elements, and indicate a fluid-like behavior of the approximations, even in the case of large data, so long as the solution remains regular.

Galerkin Finite Element Methods for Stochastic Parabolic Partial Differential Equations

- MathematicsSIAM J. Numer. Anal.
- 2005

Optimal strong convergence error estimates in the L2 and $\dot{H}^{-1}$ norms with respect to the spatial variable are obtained and the proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem.

Semidiscrete Galerkin Approximation for a Linear Stochastic Parabolic Partial Differential Equation Driven by an Additive Noise

- Mathematics
- 2004

Abstract
We study the semidiscrete Galerkin approximation of a stochastic parabolic partial differential equation forced by an additive space-time noise. The discretization in space is done by a…

Time-Splitting Methods to Solve the Stochastic Incompressible Stokes Equation

- Computer ScienceSIAM J. Numer. Anal.
- 2012

Optimal strong convergence is shown for Chorin's time-splitting scheme in the case of solenoidal noise, while computational counterexamples show poor convergence behavior in the cases of general stochastic forcing.

On the discretization in time of parabolic stochastic partial differential equations

- Mathematics, Computer ScienceMonte Carlo Methods Appl.
- 2001

Although the author is not able in this case to compute a pathwise order of the approximation, the weaker notion of order in probability is introduced and the results of the globally Lipschitz case are generalized.

Stochastic shear thickening fluids: Strong convergence of the Galerkin approximation and the energy equality

- Mathematics
- 2012

We consider a SPDE (stochastic partial differential equation) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force. Here, the extra stress…

Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra.

- Mathematics
- 2001

We discretize a steady Navier–Stokes system on a three-dimensional polyhedron by finite-elements schemes defined on two grids. In the first step, the fully nonlinear problem is solved on a coarse…