Asymptotic behavior and time discretization analysis for the non-stationary Navier-Stokes problem

“…We will frequently use the following discrete version of the Gronwall lemmas which are used in [38][39][40][41]. Lemma 4.4 For the integer n ≥ 0, let λ, τ , and a n , b n , c n be nonnegative numbers such that…”

Section: Finite Element Galerkin Approximationmentioning

confidence: 99%

On the Euler implicit/explicit iterative scheme for the stationary Oldroyd fluid

Guo

2018

Numerical Methods Partial

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In this article, we consider the stationary Oldroyd fluid equations from the large time behavior research of the nonstationary equations. Thus, to obtain its numerical solution, we first solve the nonstationary Oldroyd fluid equations via the Euler implicit/explicit finite element method with the integral term discretized by the right‐hand rectangle rule, then increase the total time (i.e., number of time steps) to approximate the solution of the original stationary equations. Under a new uniqueness condition (A2), we prove the exponential stability of the solution pair { u ¯ , p ¯ } for the stationary equations and the almost unconditional stability of the numerical method. Furthermore, we also obtain the uniform optimal H 1 and L 2 error estimates in time integral 0 ≤ t < + ∞ . Finally, several numerical experiments are provided to verify our theoretical results.

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“…By compactness, it is possible to extract a subsequence u ν k which converges strongly in C([0, T ]; L 2 (Ω)) and weakly in L 2 (0, T ; H 1 (Ω)) as ν k → 0 when k → ∞. Recalling that u ν k = v ν k + a, these modes of convergence are sufficient to pass to the limit in (5) and get…”

Section: Remarkmentioning

confidence: 99%

“…In the case of bounded domains and homogeneous Dirichlet boundary conditions, we can cite, for instance, [4] and [6], where the analysis of the convergence of the linearized implicit Euler scheme were performed. More recently, [5] and [13] considered the fully implicit Euler scheme proving convergence and stability.…”

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confidence: 99%

On the asymptotic behaviour of the 2D Navier‐Stokes equations with Navier friction conditions towards Euler equations

Guillén-González

Planas

2009

Z Angew Math Mech

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In this paper, the two dimensional incompressible Navier-Stokes problem in a bounded domain subject to nonhomogeneous Navier friction boundary conditions is considered. We establish the existence, uniqueness, and regularity of solutions to such problem and we demonstrate convergence towards the incompressible 2D Euler equations in the inviscid limit. We also show the stability and convergence of a time discrete implicit Euler's scheme as both parameters, viscosity and time step, go to zero.We consider in this paper the incompressible Navier-Stokes equations in a bounded domain Ω ⊂ R 2 subject to nonhomogeneous Navier friction boundary conditions. The problem is to find u : Ω × (0, T ) → R 2 the fluid velocity and π : Ω × (0, T ) → R the pressure, such thatwhere ν > 0 is the viscosity coefficient; n and τ are the unit outwards normal and counterclockwise tangent vectors to the boundary ∂Ω, respectively; D(u) = 1 2 (∂ i u j + ∂ j u i ) 1≤i,j≤2 is the deformation tensor; α = α(x) is a scalar friction function; b = b(x, t) and d = d(x, t) are boundary data, d describes the flow across ∂Ω and b represents some extra friction forces at the boundary. We also assume that Ω is simply connected with smooth boundary, then the incompressibility condition for the velocity leads to the following compatibility condition on the boundary data ∂Ω d(x, t) dσ = 0 for t ∈ (0, T ).The inviscid limit, also called vanishing viscosity limit, of the incompressible Navier-Stokes equations is a classical problem in Fluid Mechanics. In the whole space case, the periodic case and more generally in domains without boundaries, the inviscid limit was performed by several authors, we can refer for instance to [3,7,14]. In the physically interesting case where there exist physical boundaries the problem of convergence is essentially open. There is a discrepancy between the Dirichlet boundary conditions imposed to the Navier-Stokes equations and the tangential boundary conditions for the Euler equations, which leads to the formation of a boundary layer.Recently, the inviscid limit of the 2D incompressible Navier-Stokes equations in a bounded domain subject to Navier friction-type boundary conditions with homogeneous boundary data, that is, with b = d = 0 in (1), was studied in [2], [8]

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Asymptotic behavior and time discretization analysis for the non-stationary Navier-Stokes problem

Cited by 20 publications

References 13 publications

On large time-stepping methods for the Cahn–Hilliard equation

On large time-stepping methods for the Cahn–Hilliard equation

On the Euler implicit/explicit iterative scheme for the stationary Oldroyd fluid

On the asymptotic behaviour of the 2D Navier‐Stokes equations with Navier friction conditions towards Euler equations

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