Kaitai Li

Two-level Stabilized Finite Element Methods for the Steady Navier–Stokes Problem

Abstract.In this article, the two-level stabilized finite element formulations of the two-dimensional steady Navier–Stokes problem are analyzed. A macroelement condition is introduced for constructing the local stabilized formulation of the steady Navier–Stokes problem. By satisfying this condition the stability of the Q1−P0 quadrilateral element and the P1−P0 triangular element are established. Moreover, the two-level stabilized finite element methods involve solving one small Navier–Stokes problem on a coarse mesh with mesh size H, a large Stokes problem for the simple two-level stabilized finite element method on a fine mesh with mesh size h=O(H2) or a large general Stokes problem for the Newton two-level stabilized finite element method on a fine mesh with mesh size h=O(|log h|1/2H3). The methods we study provide an approximate solution (uh,ph) with the convergence rate of same order as the usual stabilized finite element solution, which involves solving one large Navier–Stokes problem on a fine mesh with mesh size h. Hence, our methods can save a large amount of computational time.

Pressure projection stabilized finite element method for Navier–Stokes equations with nonlinear slip boundary conditions

In this paper, we consider the pressure projection stabilized finite element method for the Navier–Stokes equation with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with Navier–Stokes operator. The H1 and L2 error estimates for the velocity and the L2 error estimate for the pressure are obtained. Finally, the numerical results are displayed to verify the theoretical analysis.

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

Summary.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.

A NEW APPROXIMATE INERTIAL MANIFOLD AND ASSOCIATED ALGORITHM

Abstract In this article the authors propose a new approximate inertial manifold(AIM) to the Navier-Stokes equations. The solutions are in the neighborhoods of this AIM with thickness δ = o(h2k+1–E). The article aims to investigate a two grids finite element approximation based on it and give error estimates of the approximate solutionn ∥ | ( u - u h * *, p - p h * * ) ∥ | ≤ C ( h 2 k + 1 - ɛ + h * ( m + 1 ) where (h, h*) and (k, m) are coarse and fine meshes and degree of finite element subspaces, respectively. These results are much better than Standard Galerkin(SG) and nonlinear Galerkin (NG) methods. For example, for 2D NS eqs and linear element, let uh, uh, u* be the SG, NG and their approximate solutions respectively, thenn ∥ u - u h ∥ 1 ≤ C h, ∥ u - u h ∥ 1 ≤ C h 2, ∥ u - u * ∥ 1 ≤ C h 3, and h* ≈ h2 for NG, h* ≈ h3/2 for theirs.

Pressure projection stabilized finite element method for Stokes problem with nonlinear slip boundary conditions

In this paper, we consider the pressure projection stabilized finite element method for the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with the Stokes operator. The H^1 and L^2 error estimates for the velocity and the L^2 error estimate for the pressure are obtained. Finally, the numerical results are displayed to verify the theoretical analysis.

THE COUPLING OF BOUNDARY ELEMENT AND FINITE ELEMENT METHODS FOR THE EXTERIOR NONSTATIONARY NAVIER-STOKES EQUATIONS

Abstract In this paper, we represent a new numerical method for solving the nonstationary Stokes equations in an unbounded domain. The technique consists in coupling the boundary integral and finite element methods. The variational formulation and well posedness of the coupling method are obtained. The convergence and optimal estimates for the approximation solution are provided.

STOKES COUPLING METHOD FOR THE EXTERIOR FLOW PART II: WELL-POSEDNESS ANALYSIS

Abstract In this paper, we recall the Stokes coupling method for solving the exterior unsteady Navier-Stokes equations. Moreover, we derive the coupling variational formulation of the Stokes coupling problem by use of the integral representations of the solution of the Stokes equations at an infinite domain. Finally, we provide some properties of the integral operators over the artificial boundary and the well-posedness of the coupling variational formulation.

A Characteristic Projection Method for Incompressible Thermal Flow

This work deals with the computation of incompressible thermal flow under the Boussinesq hypothesis, and a characteristic projection method is proposed for this. First, characteristic temporal discretization is used to obtain an upwind scheme, then at each time step the energy equation can be decoupled from momentum equations. For the remaining Stokes problem we present an improved projection method, which can overcome the numerical boundary-layer problem of the traditional projection method. In conclusion, only three independent linear elliptic equations need to be calculated at each time step; moreover, the stiffness matrices of finite-element approximation are symmetrical, positive, and time-invariant.

Existence of global attractors for a nonlinear evolution equation in Sobolev space Hk

Abstract In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation u t =Δ u + Λ u − u 3 possesses a global attractor in Sobolev space H k for all k ≥ 0, which attracts any bounded domain of H k (Ω) in the H k -norm. This result is established by using an iteration technique and regularity estimates for linear semigroup of operator, which extends the classical result from the case k ∈ [0, 1] to the case k ∈ [0, ∞).

ASYMPTOTIC BEHAVIOR OF THE DRIFT-DIFFUSION SEMICONDUCTOR EQUATIONS

Abstract This paper is devoted to the long time behavior for the Drift-diffusion semiconductor equations. It is proved that the dynamical system has a compact, connected and maximal attractor when the mobilities are constants and generation-recombination term is the Auger model; as well as the semigroup S(t) defined by the solutions map is differential. Moreover the upper bound of Hausdorff dimension for the attractor is given.