# Finite-element-based discretizations of the incompressible Navier–Stokes equations with multiplicative random forcing

@article{Brzeniak2013FiniteelementbasedDO, title={Finite-element-based discretizations of the incompressible Navier–Stokes equations with multiplicative random forcing}, author={Zdzisław Brzeźniak and Erich Carelli and Andreas Prohl}, journal={Ima Journal of Numerical Analysis}, year={2013}, volume={33}, pages={771-824} }

We study finite element based space-time discretisations of the incompressible Navier-Stokes equations with noise. In three dimensions, sequences of numerical solutions construct weak martingale solutions for vanishing discretisation parameters. In the two dimensional case, numerical solutions converge to the unique strong solution.

## 61 Citations

Time-discretization of stochastic 2-D Navier-Stokes equations with a penalty-projection method

- Computer ScienceNumerische Mathematik
- 2019

A time-discretization of the stochastic incompressible Navier--Stokes problem by penalty method is analyzed and the law of total probability is used to obtain the strong convergence of the scheme for both variables.

Convergent finite element based discretization of a stochastic two‐phase flow model

- Mathematics, Computer ScienceZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
- 2021

It is proved that the proposed numerical scheme is unconditionally solvable, has finite energies and constructs weak martingale solutions of the stochastic Allen‐Cahn‐Navier‐Stokes system when the discretisation step tends to zero.

Numerical approximation of the stochastic Navier-Stokes equations through artificial compressibility

- MathematicsArXiv
- 2022

A constructive numerical approximation of the two-dimensional unsteady stochastic Navier-Stokes equations of an incompressible ﬂuid is proposed via a pseudo-compressibility technique involving a…

Numerical Analysis of Fully discrete Finite element Methods for the stochastic Navier-Stokes Equations with Multiplicative Noise

- Mathematics
- 2021

Fully discrete finite element approximation of the stochastic Cahn–Hilliard–Navier–Stokes system

- Mathematics, Computer Science
- 2020

It is proved that the proposed numerical scheme satisfies the discrete mass conservative law, has finite energies and constructs a weak martingale solution of the stochastic Cahn–Hilliard–Navier–Stokes system when the discretization step tends to zero.

Layer methods for stochastic Navier-Stokes equations using simplest characteristics

- MathematicsJ. Comput. Appl. Math.
- 2016

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.

Layer methods for Navier-Stokes equations with additive noise

- Mathematics
- 2013

We propose and study a number of layer methods for stochastic Navier-Stokes equations (SNSE) with spatial periodic boundary conditions and additive noise. The methods are constructed using…

Approximation of deterministic and stochastic Navier-Stokes equations in vorticity-velocity formulation

- Mathematics
- 2018

We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions in the vorticity-velocity formulation. The approximation is based on freezing the…

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.

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