A TWO-LEVEL FINITE ELEMENT GALERKIN METHOD FOR THE NONSTATIONARY NAVIER-STOKES EQUATIONS II： TIME DISCRETIZATION

In this article we consider the fully discrete two-level finite element Galerkin method for the two-dimensional nonstationary incompressible Navier-Stokes equations. This method consists in dealing with the fully discrete nonlinear Navier-Stokes problem on a coarse mesh with width $H$ and the fully discrete linear generalized Stokes problem on a fine mesh with width $h << H$. Our results show that if we choose $H=O(h^{1/2}$) this method is as the same stability and convergence as the fully discrete standard finite element Galerkin method which needs dealing with the fully discrete nonlinear Navier-Stokes problem on a fine mesh with width $h$. However, our method is cheaper than the standard fully discrete finite element Galerkin method.     

Reduced Finite Element Discretizations of the Stokes and Navier-Stokes Equations

ABSTRACT

If finite element spaces for the velocity and pressure do not satisfy the Babu?ka-Brezzi condition, a stable conforming discretization of the Stokes or Navier-Stokes equations can be obtained by enriching the velocity space by suitable functions. Writing any function from the enriched space as a sum of a function from the original space and a function from the supplementary space, the discretization will contain a number of additional terms compared with a conforming discretization for the original pair of spaces. We show that not all these terms are necessary for the solvability of the discrete problem and for optimal convergence properties of the discrete solutions, which is useful for saving computer memory and for establishing a connection to stabilized methods.     

A Study of Multiple Solutions for the Navier-Stokes Equations by a Finite Element Method

In this paper, a finite element method is proposed to investigate multiple solutions of the Navier-Stokes equations for an unsteady, laminar, incompressible flow in a porous expanding channel. Dual or triple solutions for the fixed values of the wall suction Reynolds number $R$ and the expansion ratio $α$ are obtained numerically. The computed multiple solutions for the symmetric flow are validated by comparing them with approximate analytic solutions obtained by the similarity transformation and homotopy analysis method. Unlike previous works, our method deals with the Navier-Stokes equations directly and thus has no similarity and other restrictions as in previous works. Finally we use the method to study multiple solutions for three cases of the asymmetric flow (which has not been studied before using the similarity-type techniques).     

A TWO-LEVEL FINITE ELEMENT GALERKIN METHOD FOR THE NONSTATIONARY NAVIER-STOKES EQUATIONS I： SPATIAL DISCRETIZATION

In this article we consider a two-level finite element Galerkin method using mixed finite elements for the two-dimensional nonstationary incompressible Navier-Stokes equations. The method yields a $H^1$-optimal velocity approximation and a $L_2$-optimal pressure approximation. The two-level finite element Galerkin method involves solving one small, nonlinear Navier-Stokes problem on the coarse mesh with mesh size $H$, one linear Stokes problem on the fine mesh with mesh size $h << H$. The algorithm we study produces an approximate solution with the optimal, asymptotic in $h$, accuracy.     

Long-Time Turbulence Model Deduced from the Navier-Stokes Equations

The author shows the existence of long-time averages to turbulent solutions of the Navier-Stokes equations and determines the equations satisfied by them, involving a Reynolds stress that is shown to be dissipative.     

A Variational Analysis for the Moving Finite Element Method for Gradient Flows

By using the Onsager principle as an approximation tool, we give a novel derivation for the moving finite element method for gradient flow equations. We show that the discretized problem has the same energy dissipation structure as the continuous one. This enables us to do numerical analysis for the stationary solution of a nonlinear reaction diffusion equation using the approximation theory of free-knot piecewise polynomials. We show that under certain conditions the solution obtained by the moving finite element method converges to a local minimizer of the total energy when time goes to infinity. The global minimizer, once it is detected by the discrete scheme, approximates the continuous stationary solution in optimal order. Numerical examples for a linear diffusion equation and a nonlinear Allen-Cahn equation are given to verify the analytical results.     

On Some Model Equations of Euler and Navier-Stokes Equations

The author proposes a two-dimensional generalization of Constantin-Lax-Majda model. Some results about singular solutions are given. This model might be the first step toward the singular solutions of the Euler equations. Along the same line(vorticity formulation), the author presents some further model equations. He possibly models various aspects of difficulties related with the singular solutions of the Euler and Navier-Stokes equations. Some discussions on the possible connection between tur...     

Pressure Boundary Conditions for Blood Flows

Simulations of blood flows in arteries require numerical solutions of fluid-structure interactions involving Navier-Stokes equations coupled with large displacement visco-elasticity for the vessels. Among the various simplifications which have been proposed, the surface pressure model leads to a hierarchy of simpler models including one that involves only the pressure. The model exhibits fundamental frequencies which can be computed and compared with the pulse. Yet unconditionally stable time discretizations can be constructed by combining implicit time schemes with Galerkin-characteristic discretization of the convection terms in the Navier-Stokes equations. Such problems with prescribed pressure on the walls will be shown to be efficient and accurate as an approximation of the full fluid structure interaction problem.     

On The Finite Element Approximation of Variational Inequalities with Noncoercive Operators

In this article, we introduce a new method to analyze the convergence of the standard finite element method for variational inequalities with noncoercive operators. We derive an optimal L ^∞ error estimate by combining the Bensoussan-Lions algorithm with the concept of subsolutions.     

定常Navier—Stokes方程的Petrov—Galerkin方法

研究了定常Navier－Stokes方程的四种Petrov－Galerkin有限元方法：PG1，PG2，SD和GLS．它们都是稳定的，避免了经典混合方法中必要的Babuska－Brezzi条件．给出了各种方法有限元解的存在性、唯一性和唯一解的误差估计．     

对流扩散方程的有限体积-有限元方法的误差估计

本文结合有限体积方法和有限元方法处理非线性对流扩散问题,非线性对流项利用有限体积方法处理,扩散项利用有限元方法离散,并给近似解的误差估计。     

A Second Order Nonconforming Triangular Mixed Finite Element Scheme for the Stationary Navier-Stokes Equations

In this paper,a nonconforming triangular mixed finite element scheme with second order convergence behavior is proposed for the stationary Navier-Stokes equations.The new nonconforming triangular element is taken as approximation space for the velocity and the linear element for the pressure.The convergence analysis is presented and optimal error estimates of both broken H1-norm and L2-norm for velocity as well as the L2-norm for the pressure are derived.     

A Logarithmically Improved Blow-up Criterion for a Simplified Ericksen-Leslie System Modeling the Liquid Crystal Flows

In this paper, we prove a logarithmically improved blow-up criterion in terms of the homogeneous Besov spaces for a simplified 3D Ericksen-Leslie system modeling the hydrodynamic flow of nematic liquid crystal. The result shows that if a local smooth solution (u,d) satisfies $$∫^T_0\frac{||u||^{\frac{2}{1-r}}_{\dot{B}^{-r}{∞,∞}}+||∇ d||²_{L^∞}}{1+1n(e+||u||_H^S+||∇ d||_H^S)}dt‹∞$$ with 0 ≤ r ‹ 1 and s ≥ 3, then the solution (u,d) can be smoothly extended beyond the time T.     

A Two‐Parameter Stabilized Finite Element Method for Incompressible Flows

Based on two‐grid discretizations, a two‐parameter stabilized finite element method for the steady incompressible Navier–Stokes equations at high Reynolds numbers is presented and studied. In this method, a stabilized Navier–Stokes problem is first solved on a coarse grid, and then a correction is calculated on a fine grid by solving a stabilized linear problem. The stabilization term for the nonlinear Navier–Stokes equations on the coarse grid is based on an elliptic projection, which projects higher‐order finite element interpolants of the velocity into a lower‐order finite element interpolation space. For the linear problem on the fine grid, either the same stabilization approach (with a different stabilization parameter) as that for the coarse grid problem or a completely different stabilization approach could be employed. Error bounds for the discrete solutions are estimated. Algorithmic parameter scalings of the method are also derived. The theoretical results show that, with suitable scalings of the algorithmic parameters, this method can yield an optimal convergence rate. Numerical results are provided to verify the theoretical predictions and demonstrate the effectiveness of the proposed method. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 425–444, 2017     

定常Satokes方程的窄边四边形有限元法

本文提出了一个试探函数不满足divν＝0的定常Stokes方程的窄边四边形有限元法，利用窄边四边形等参有限元插值定理和Falk的思想，得到了H^1(Ω）模误差的最优收敛估计阶O（h）。     

A Characteristic Finite Volume Element Method for the Air Pollution Model

In this article, a characteristic finite volume element method is presented for solving air pollution models. The convection term is discretized using the characteristic method and diffusion term is approximated by finite volume element method. Compared with standard finite volume element method, our proposed method is more accurate and efficient, especially suitable to solve convection-dominated problems. The proposed numerical schemes are analyzed for convergence in L ₂ norm. Some numerical results are presented to demonstrate the efficiency and accuracy of the method.     

一类非线性波动方程的非协调有限元方法

在半离散格式下,研究了一类非线性波动方程的非协调有限元逼近.首先证明了该格式解的存在性和唯一性,给出了稳定性分析和误差分析,其次得到了最优的误差估计.     

Euler-Lagrange Simulations of Turbulent Two-phase Flows with an Advanced Deterministic Bubble Coalescence Model

The present paper is concerned with an advanced version of the deterministic coalescence model [1] and its application to turbulent bubble-laden flows. The major drawback of this film drainage model – its incapability to handle large numbers of coalescence processes – is avoided by an a-priori determination of a parametric relation for the transition time between the main drainage mechanisms. The comparison with experimental results yields a good overall agreement, while the application to a downward pipe flow demonstrates the ability of the model to efficiently handle large numbers of coalescence processes. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

求解超声速无粘绕流的杂交型有限元格式

本文从一维双曲型标量方程出发,以一个普通二阶有限元格式及由其单边对角化导出的一阶单调型格式为基础,构造出一种具有单调性的杂交型有限元格式.为了向二维Euler方程组情形推广,所采用的开关函数是基于流场梯度的局部函数,并专门考虑了相邻单元的影响.二维情形的算例表明新格式可以明显抑制激波附近的振荡.