Time-dependent Navier-Stokes Problem Research Articles

An algorithm for solving the Navier–Stokes problem with mixed boundary conditions

We consider a time-dependent Navier–Stokes problem in dimension two and three provided with mixed boundary conditions. We propose an iterative algorithm and its implementation for resolving this considered problem. The discretization is based on a backward Euler scheme with respect to the time variable and the spectral method with respect to the space variables. We present some numerical experiments which confirm the interest of the discretization.

Spectral discretization of the time-dependent Navier-Stokes problem with mixed boundary conditions

Abstract In this work, we handle a time-dependent Navier-Stokes problem in dimension three with a mixed boundary conditions. The variational formulation is written considering three independent unknowns: vorticity, velocity, and pressure. We use the backward Euler scheme for time discretization and the spectral method for space discretization. We present a complete numerical analysis linked to this variational formulation, which leads us to a priori error estimate.

Superconvergent analysis of a nonconforming mixed finite element method for time-dependent damped Navier–Stokes equations

The main aim of this paper is to develop a nonconforming mixed finite element method for the time-dependent Navier–Stokes problem with nonlinear damping term. The superconvergent analysis of a backward Euler fully-discrete scheme is presented, where the constrained nonconforming rotated (CNR) $$Q_1$$ element and the $$Q_0$$ element are used to approximate the velocity $${\varvec{u}}$$ and the pressure p, respectively. By use of the characters of the element pair together with some striking skills, i.e., mean-value skill and a new transforming skill with respect to $$\tau $$ , the superclose estimates of $$O(h^2+\tau )$$ for $${\varvec{u}}$$ in broken $$H^1$$ -norm and p in $$L^2$$ -norm are deduced rigorously. Furthermore, the global superconvergent results are obtained through the interpolated postprocessing technique. Finally, some numerical results are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and $$\tau $$ , the time step.

Long time unconditional stability of a two-level hybrid method for nonstationary incompressible Navier–Stokes equations

This paper presents a two-level MacCormack rapid solver method for solving a 2D time dependent Navier–Stokes problem that models an incompressible fluid flow. A decoupling approach based on the coupling term via spatial extrapolation is proposed for devising decoupled marching algorithms for the nonstationary incompressible model. In this hybrid method, an explicit MacCormack scheme provides a H 1 -optimal velocity approximation and a L 2 -optimal pressure approximation by solving a nonlinear Navier–Stokes problem on a coarse grid with mesh size H , while an implicit Crank–Nicolson algorithm consists in dealing with the fully discrete linear generalized Stokes problem on a fine grid with mesh size h ≪ H . The theoretical result suggests that our method is unconditionally stable over long time intervals. While the numerical evidences both confirm the theoretic analysis and show that the algorithm is convergent, the tests also suggest that our method is both cheaper and faster than the two-level finite element Galerkin scheme.

A posteriori error analysis of the time dependent Navier–Stokes equations with mixed boundary conditions

In this paper we study the time dependent Navier–Stokes problem with mixed boundary conditions. The problem is discretized by the backward Euler’s scheme in time and finite elements in space. We establish optimal a posteriori error estimates with two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization. We finish with numerical validation experiments.

Two-grid finite element scheme for the fully discrete time-dependent Navier–Stokes problem

In this Note, we study a two-grid scheme fully discrete in time and space for solving the Navier–Stokes system. In the first step, the fully non-linear problem is discretized in space on a coarse grid with mesh-size H and time step k. In the second step, the problem is discretized in space on a fine grid with mesh-size h and the same time step, and linearized around the velocity u H computed in the first step. The two-grid strategy is motivated by the fact that under suitable assumptions, the contribution of u H to the error in the non-linear term, is measured in the L 2 norm in space and time, and thus has hopefully a higher-order than if it were measured in the H 1 norm in space. We present the following results: if h = H 2 = k , then the global error of the two-grid algorithm is of the order of h, the same as would have been obtained if the non-linear problem had been solved directly on the fine grid. To cite this article: H. Abboud et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).

A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem

A fully discrete stabilized finite-element method is presented for the two-dimensional time-dependent Navier-Stokes problem. The spatial discretization is based on a finite-element space pair (X h , M h ) for the approximation of the velocity and the pressure, constructed by using the Q 1 - P 0 quadrilateral element or the P 1 - P 0 triangular element; the time discretization is based on the Euler semi-implicit scheme. It is shown that the proposed fully discrete stabilized finite-element method results in the optimal order error bounds for the velocity and the pressure.

Nonhomogeneous Dirichlet Navier—Stokes Problems in Low Regularity Lp Sobolev Spaces

The time-dependent Navier—Stokes problem on an interior or exterior smooth domain, with nonhomogeneous Dirichlet boundary condition, is treated in anisotropic L p Sobolev spaces (1 < p < $ \infty $ ) of Bessel-potential type $ H^{s+2,s/2+1}_p $ or Besov type $ B^{s+2,s/2+1}_p $ by use of a reformulation of the linearized problem to a parabolic pseudodifferential boundary value problem. Earlier studies required $ s > {\frac 1p} - 1 $ ; the present work extends the solvability to spaces with $ s > {\frac 1p} - 2 $ for zero initial data ( $ s > - 2 $ if f = 0), $ s > {\frac 2p} - 2 $ for nonzero initial data, with s,p subject to other conditions stemming from the nonlinearity.