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

Brzeźniak, Zdzisław; Carelli, Erich; Prohl, Andreas

doi:10.1093/imanum/drs032

Cited by 76 publications

(88 citation statements)

References 20 publications

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“…Remark 5. (a) We note that a similar estimate to (4.11) was proved in [9] but (4.12) seems new. We also emphasize that the above stability estimates will not be used in our error analysis, instead, those given in Lemma 3.1 will be crucially used.…”

Section: 2supporting

confidence: 70%

A Fully Discrete Mixed Finite Element Method for the Stochastic Cahn–Hilliard Equation with Gradient-Type Multiplicative Noise

Feng

Zhang

2020

J Sci Comput

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This paper is concerned with fully discrete mixed finite element approximations of the time-dependent stochastic Stokes equations with multiplicative noise. A prototypical method, which comprises of the Euler-Maruyama scheme for time discretization and the Taylor-Hood mixed element for spatial discretization is studied in detail. Strong convergence with rates is established not only for the velocity approximation but also for the pressure approximation (in a time-averaged fashion). A stochastic inf-sup condition is established and used in a nonstandard way to obtain the error estimate for the pressure approximation in the time-averaged fashion. Numerical results are also provided to validate the theoretical results and to gauge the performance of the proposed fully discrete mixed finite methods.

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Section: 2supporting

confidence: 70%

A Fully Discrete Mixed Finite Element Method for the Stochastic Cahn–Hilliard Equation with Gradient-Type Multiplicative Noise

Feng

Zhang

2020

J Sci Comput

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“…As further related work on numerical approximation of SPDEs without a global monotonicity assumption we mention the pathwise convergence of a spectral Galerkin method for the stochastic Burgers equation studied in [2,3], while for the same equation convergence in probability is established in [26] for the Backward Euler method. The stochastic Navier-Stokes equation is considered in [5,7], in particular, in [7] the authors obtain a result similar to our Theorem 5.5 (stated in a slightly different form). Finally, we mention the recent work [17], where strong convergence is proved, without rate, for a spectral nonlinearity-truncated accelerated exponential Euler-type approximation for the stochastic Kuramoto-Sivashinsky equation driven by space-time white noise in spatial dimension d = 1, an equation rather similar in structure to the Cahn-Hilliard-Cook equation.…”

Section: Introductionsupporting

confidence: 65%

Strong Convergence of a Fully Discrete Finite Element Approximation of the Stochastic Cahn--Hilliard Equation

Furihata¹,

Kovács²,

Larsson³

et al. 2018

SIAM J. Numer. Anal.

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We consider the stochastic Cahn-Hilliard equation driven by additive Gaussian noise in a convex domain with polygonal boundary in dimension d ≤ 3. We discretize the equation using a standard finite element method in space and a fully implicit backward Euler method in time. By proving optimal error estimates on subsets of the probability space with arbitrarily large probability and uniform-in-time moment bounds we show that the numerical solution converges strongly to the solution as the discretization parameters tend to zero.2000 Mathematics Subject Classification. 60H15, 60H35, 65C30.

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“…Let us recall Lemma 3.1 in [9], which proves moment estimates of the solution to (4.1). Note that here only dyadic moments are computed because of the induction argument which relates two consecutive dyadic numbers (see step 4 of the proof of Lemma 3.1 in [8]).…”

Section: Euler Time Schemesmentioning

confidence: 99%

“…The stochastic Navier-Stokes equations with a multiplicative noise (1.1) have been investigated by Z. Brzezniak, E. Carelli and A. Prohl in [8]. There, space discretization based on finite elements and Euler schemes for the time discretization have been implemented.…”

Section: Introductionmentioning

confidence: 99%

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

Bessaih

Millet

2018

IMA Journal of Numerical Analysis

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We prove that some time discretization schemes for the 2D Navier-Stokes equations on the torus subject to a random perturbation converge in L 2 (Ω). 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 fully implicit and semi-implicit time Euler discretizations, and of a splitting scheme. The speed of the L 2 (Ω)-convergence depends on the diffusion coefficient and on the viscosity parameter.2000 Mathematics Subject Classification. Primary 60H15, 60H35; Secondary 76D06, 76M35.

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

Cited by 76 publications

References 20 publications

A Fully Discrete Mixed Finite Element Method for the Stochastic Cahn–Hilliard Equation with Gradient-Type Multiplicative Noise

A Fully Discrete Mixed Finite Element Method for the Stochastic Cahn–Hilliard Equation with Gradient-Type Multiplicative Noise

Strong Convergence of a Fully Discrete Finite Element Approximation of the Stochastic Cahn--Hilliard Equation

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

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