Nonstationary Navier-Stokes Problem Research Articles

The Optimal Error Estimate of the Fully Discrete Locally Stabilized Finite Volume Method for the Non-Stationary Navier-Stokes Problem.

This paper proves the optimal estimations of a low-order spatial-temporal fully discrete method for the non-stationary Navier-Stokes Problem. In this paper, the semi-implicit scheme based on Euler method is adopted for time discretization, while the special finite volume scheme is adopted for space discretization. Specifically, the spatial discretization adopts the traditional triangle trial function pair, combined with macro element form to ensure local stability. The theoretical analysis results show that under certain conditions, the full discretization proposed here has the characteristics of local stability, and we can indeed obtain the optimal theoretic and numerical order error estimation of velocity and pressure. This helps to enrich the corresponding theoretical results.

Global convergence in the strong norm of an iterative method for the nonstationary Navier-Stokes problem

It is established below that the existence of a strong solution to the nonlinear Navier–Stokes problem with any initial approximation from the class of strong solu� tions is equivalent to the global convergence of the modified iteration sequence to the sought solution in the norm of the class of strong solutions at a conver� gence rate higher than a geometric progression rate with a ratio arbitrarily close to zero. Interestingly, the sought strong solution is not involved in the construc� tion of the modified iteration sequence and, in fact, the global boundedness of this sequence in the norm of the class of strong solutions becomes a criterion for the global solvability of the nonlinear Navier–Stokes problem in the class of strong solutions (for more detail, see Section 2 below).

On the global convergence rate of an iterative method for the Navier-Stokes problem

For the nonlinear nonstationary Navier–Stokes problem, we consider an iterative method in which each iteration step involves solving an initial–bound� ary value problem with convective terms linearized on the solution obtained at the previous iteration. Assum� ing that the sought weak solution belongs to the Ladyzhenskaya–Prodi–Serrin class [1, 2], estimates are obtained that guarantee the fast global conver� gence of the constructed successive approximations in the norm of the class of Hopf weak solutions irrespec� tive of the choice of the initial approximation. More specifically, for any initial approximation, the global convergence to the sought solution is proved to have a geometric progression rate with a ratio arbitrarily close to zero. In a bounded domain Ω ⊂ 3 with a smooth boundary ∂Ω ∈ C 2 , consider the Navier–Stokes prob� lem, i.e., the initial–boundary value problem for the nonlinear nonstationary Navier–Stokes equations

On dispersive effect of the Coriolis force for the stationary Navier–Stokes equations

The dispersive effect of the Coriolis force for the stationary and non-stationary Navier–Stokes equations is investigated. Existence of a unique solution is shown for arbitrary large external force provided the Coriolis force is large enough. In addition to the stationary case, counterparts of several classical results for the non-stationary Navier–Stokes problem have been proven. The analysis is carried out in a new framework of the Fourier–Besov spaces.

Finite volume method based on stabilized finite elements for the nonstationary Navier–Stokes problem

AbstractA finite volume method based on stabilized finite element for the two‐dimensional nonstationary Navier–Stokes equations is investigated in this work. As in stabilized finite element method, macroelement condition is introduced for constructing the local stabilized formulation of the nonstationary Navier–Stokes equations. Moreover, for P1 − P0 element, the H1 error estimate of optimal order for finite volume solution (uh,ph) is analyzed. And, a uniform H1 error estimate of optimal order for finite volume solution (uh, ph) is also obtained if the uniqueness condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Multi-level spectral Galerkin method for the Navier–Stokes equations, II: time discretization

A fully discrete multi-level spectral Galerkin method in space–time for the two-dimensional nonstationary Navier–Stokes problem is considered. The method is a multi-scale method in which the fully nonlinear Navier–Stokes problem is only solved on the lowest-dimensional space $H_{m_{1}}$ with the largest time step Δt 1; subsequent approximations are generated on a succession of higher-dimensional spaces $H_{m_{j}}$ with small time step Δt j by solving a linearized Navier–Stokes problem about the solution on the previous level. Some error estimates are also presented for the J-level spectral Galerkin method. The scaling relations of the dimensional numbers and time mesh widths that lead to optimal accuracy of the approximate solution in H 1-norm and L 2-norm are investigated, i.e., m j∼m j−1 3/2 , Δt j∼Δt j−1 3/2 , j=2,. . .,J. We demonstrate theoretically that a fully discrete J-level spectral Galerkin method is significantly more efficient than the standard one-level spectral Galerkin method.

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

The asymptotic behavior and the Euler time discretization analysis are presented for the two-dimensional non-stationary Navier-Stokes problem. If the data ν and f(t) satisfy a uniqueness condition corresponding to the stationary Navier-Stokes problem, we then obtain the convergence of the non-stationary Navier-Stokes problem to the stationary Navier-Stokes problem and the uniform boundedness, stability and error estimates of the Euler time discretization for the non-stationary Navier-Stokes problem.

Coupling boundary integral and finite element methods for the Oseen coupled problem

In this paper, we represent an Oseen coupled problem and related numerical method for solving the nonstationary Navier-Stokes problem in an unbounded domain. The Oseen coupled problem consists of a coupling between the Navier-Stokes problem in an inner region and Oseen problem in an outer region. The related numerical method consists of coupling the boundary integral and the finite element method to solve the coupled problem. The variational formulation of the coupled problem and its well posedness are obtained. The optimal error estimates between the numerical solution of the Oseen coupled problem and the exact solution of the Navier-Stokes problem are provided.

MAXIMUM NORM ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS OF THE STATIONARY AND NONSTATIONARY NAVIER-STOKES PROBLEMS

This paper deals with maximum norm error estimates of conforming finite element approximate solutions for the stationary and nonstationary Navier-Stokes problems in a plane bounded domain, using the so-called velocity-pressure mixed variational formulation. Quasi-optimal maximum norm error estimates of the velocity and its first derivatives, and the pressure are obtained for conforming finite element approximations of the stationary Navier-Stokes problem by some estimates of the Green function and their finite element approximations for Stokes problem. Moreover, with the method of Navier-Stokes projection, quasi-optimal maximum norm error estimate results are shown for semidiscrete conforming finite element approximations of the nonstationary Navier-Stokes problem.

Finite Element Approximation of the Nonstationary Navier–Stokes Problem III. Smoothing Property and Higher Order Error Estimates for Spatial Discretization

This paper continues our error analysis of finite element Galerkin approximation of the nonstationary Navier–Stokes equations. Optimal order error estimates, both local and global, are derived for higher order finite elements under appropriate assumptions about the smoothness and stability of the solution. These assumptions take into account the loss of regularity at $t = 0$ that one generally has to expect in the absence of higher order nonlocal compatibility conditions for the data of the problem.

Finite Element Approximation of the Nonstationary Navier–Stokes Problem, Part II: Stability of Solutions and Error Estimates Uniform in Time

In this paper, assumptions about the stability of a solution are introduced in the numerical analysis of the nonstationary Navier–Stokes problem, for the purpose of extending local a priori error estimates, and local a posteriori error estimates, globally in time.

Finite Element Approximation of the Nonstationary Navier–Stokes Problem. I. Regularity of Solutions and Second-Order Error Estimates for Spatial Discretization

This is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary Navier–Stokes equations. Second-order error estimates are proven for spatial discretization, using conforming or nonconforming elements. The results indicate a fluid-like behavior of the approximations, even in the case of large data, so long as the solution remains regular. The analysis is based on sharp a priori estimates for the solution, particularly reflecting its behavior as$t \to 0$ and as $t \to \infty $. It is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the Navier–Stokes equations, as it depends on nonlocal compatibility conditions for the data. The results which are presented here are independent of such compatibility conditions, which cannot be verified in practice.

A difference-scheme with error O(?2 + �h�2) for a Navier-Stokes system

A system of quasilinear equations of parabolic type which approximates a nonstationary Navier-Stokes problem is considered in this article. Triple layered implicit difference schemes with a linear operator on the upper layer are constructed for this system. Rapidly converging iterative methods can be applied to find a solution on the upper layer. It is proved that the proposed scheme has error O(τ2+¦h¦2).