定常Navier-Stokes方程的一种高效稳定有限元方法

提出了二维定常Navier-Stokes(N-S)方程的一种两层稳定有限元方法.该方法基于局部高斯积分技术,通过不满足inf-sup条件的低次等阶有限元对N-S方程进行有限元求解.该方法在粗网格上解定常N-S方程,在细网格上只需解一个Stokes方程.误差分析和数值试验都表明:两层稳定有限元方法与直接在细网格上采用的传统有限元方法得到的解具有同阶的收敛性,但两层稳定有限元方法节省了大量的工作时间.     

求解定常不可压Stokes方程的两层罚函数方法

借助于两套有限元网格空间提出了一种求解定常不可压Stokes方程的两层罚函数方法.该方法只需要求解粗网格空间上的Stokes方程和细网格空间上的两个易于求解的罚参数方程（离散后的线性方程组具有相同的对称正定系数矩阵）.收敛性分析表明粗网格空间相对于细网格空间可以选择很小，并且罚参数的选取只与粗网格步长和问题的正则性有关.因此罚参数不必选择很小仍能够得到最优解.最后通过数值算例验证了上述理论结果，并且数值对比可知两层罚函数方法对于求解定常不可压Stokes方程具有很好的效果.     

非定常不可压Navier-Stokes方程两重网格方法的稳定性和L2误差估计

讨论了二维非定常不可压Navier-Stokes方程的两重网格方法．此方法包括在粗网格上求解一个非线性问题,在细网格上求解一个Stokes问题．采用一种新的全离散(时间离散用Crank-Nicolson格式,空间离散用混合有限元方法)格式数值求解N-S方程．证明了该全离散格式的稳定性．给出了L2误差估计．对比标准有限元方法,在保持同样精度的前提下,TGM能节省大量的计算量．     

一类非线性对流扩散方程两重网格特征有限元方法及误差估计

主要研究了一类非线性对流扩散方程的全离散特征有限元方法的两重网格算法及其误差估计.首先在网格步长为H的粗网格上计算一个较小的非线性问题,然后利用一阶牛顿迭代和粗网格解将网格步长为h的细网格上的非线性问题转化为线性问题求解.由于非线性问题的求解仅在粗网格上进行,该两重网格算法可以节省大量的计算工作量,同时具有较高的精度,证明了该两重网格算法L~2模先验误差估计结果为O(△t+h~2+H~(4-d/2)),其中d为空间维数.     

一类Stokes方程的最小二乘混合元方法

对定常和非定常两种类型的Stokes方程建立了一类新的最小二乘混合元方法,并进行了分析,对定常的方程,采用了对u和σ的不同指标的有限元空间进行计算(LBB条件不需要),得到了最优的H1和L2模估计.对非定常的方程,采用了传统的Raviart-Thomas混合元空间,得到了最优的L2模估计.     

拟Newton流的一种新的稳定化方法

对一般的拟Newton流问题,针对(双)线性/(双)线性和(双)线性/常数两种低阶有限元空间,提出了一种新的稳定化方法．该方法可以看成压力投影稳定化方法从Stokes问题到拟Newton流问题的推广与发展．在速度属于W1,r(Ω),压力属于Lr′(Ω)(1/r+1/r′=1)下,给出了误差估计．服从幂律及Carreau分布的拟Newton流问题可看成该文的特殊情况．进一步地,还给出了基于残差的后验误差估计．最后给出的数值算例验证了理论结果．     

平面弹性问题自适应有限元方法的收敛性分析

针对平面弹性问题,首先采用基于最新顶点二分法的网格加密方法,给出一种不需要标记振荡项和加密单元、不需要满足"内节点"性质的自适应有限元方法.其次,通过对各层网格上解函数和误差指示子的分析,利用相邻网格层上解函数的正交性、解函数和真解函数的能量误差的上界估计、相邻网格层上误差指示子的近似压缩性等结果,从理论上严格证明了该自适应有限元方法是收敛的.最后数值实验验证了该自适应有限元方法是收敛的和鲁棒的.     

非定常Stokes方程的全离散稳定化有限元格式

本文研究二维非定常Stokes方程全离散稳定化有限元方法.首先给出关于时间向后一步Euler半离散格式,然后直接从该时间半离散格式出发,构造基于两局部高斯积分的稳定化全离散有限元格式,其中空间用P_1—P_1元逼近,证明有限元解的误差估计.本文的研究方法使得理论证明变得更加简便,也是处理非定常Stokes方程的一种新的途径.     

外部非定常Navier-Stokes方程的非线性Galerkin算法

提出了求解外部非定常Navier-stokes方程的有限元边界元耦合的非线性Galerkin算法,证明了相应变分问题的正则性和数值解的收敛速度。收敛性分析表明如果选取粗网格尺度H是细网格尺度h的开平方数量级,则该算法提供了与古典Galerkin算法同阶的收敛速度。然而非线性Galerkin算法仅仅需要在粗网格解非线性问题,在细网格上解线性问题。因此,该算法可以节省计算工作量。     

Oseen方程的一种新的局部投影稳定化方法

对Oseen方程提出一种新的局部投影稳定化有限元方法,并且速度和压力采用inf-sup稳定的非协调有限元空间逼近．局部投影稳定化项仅加在子网格上(H≥h);与RFB方法相比,该方法稳定性项简单,并且可以克服对流占优．最后,通过实验证明,数值结果和理论结果完全一致．     

Acceleration of two-grid stabilized mixed finite element method for the Stokes eigenvalue problem

This paper provides an accelerated two-grid stabilized mixed finite element scheme for the Stokes eigenvalue problem based on the pressure projection. With the scheme, the solution of the Stokes eigenvalue problem on a fine grid is reduced to the solution of the Stokes eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid. By solving a slightly different linear problem on the fine grid, the new algorithm significantly improves the theoretical error estimate which allows a much coarser mesh to achieve the same asymptotic convergence rate. Finally, numerical experiments are shown to verify the high efficiency and the theoretical results of the new method.     

Two-level stabilized nonconforming finite element method for the Stokes equations

In this article, we present a new two-level stabilized nonconforming finite elements method for the two dimensional Stokes problem. This method is based on a local Gauss integration technique and the mixed nonconforming finite element of the NCP ₁ — P ₁ pair (nonconforming linear element for the velocity, conforming linear element for the pressure). The two-level stabilized finite element method involves solving a small stabilized Stokes problem on a coarse mesh with mesh size H and a large stabilized Stokes problem on a fine mesh size h = H/3. Numerical results are presented to show the convergence performance of this combined algorithm.     

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.     

Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems

In this paper we propose a method for improving the convergence rate of the mixed finite element approximations for the Stokes eigenvalue problem. It is based on a postprocessing strategy that consists of solving an additional Stokes source problem on an augmented mixed finite element space which can be constructed either by refining the mesh or by using the same mesh but increasing the order of the mixed finite element space. Dedicated to Ivan Hlaváček on the occasion of his 75th birthday     

Navier-Stokes方程带Backtracking技巧的两重网格算法

1 引 言考虑二维不可压 Navier-Stokes方程:     

A multilevel correction method for Stokes eigenvalue problems and its applications

In this paper, a new multilevel correction scheme is proposed to solve Stokes eigenvalue problems by the finite element method. This new scheme contains a series of correction steps, and the accuracy of eigenpair approximation can be improved after each step. In each correction step, we only need to solve a Stokes problem on the corresponding fine finite element space and a Stokes eigenvalue problem on the coarsest finite element space. This correction scheme can improve the efficiency of solving Stokes eigenvalue problems by the finite element method. As applications of this multilevel correction method, a multigrid method and an adaptive finite element technique are introduced for Stokes eigenvalue problems. Some numerical results are given to validate our schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

Two-level pressure projection finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions

The two-level pressure projection stabilized finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions are investigated in this paper, whose variational formulation is the Navier-Stokes type variational inequality problem of the second kind. Based on the P₁-P₁ triangular element and using the pressure projection stabilized finite element method, we solve a small Navier-Stokes type variational inequality problem on the coarse mesh with mesh size H and solve a large Stokes type variational inequality problem for simple iteration or a large Oseen type variational inequality problem for Oseen iteration on the fine mesh with mesh size h. The error analysis obtained in this paper shows that if h=O(H²), the two-level stabilized methods have the same convergence orders as the usual one-level stabilized finite element methods, which is only solving a large Navier-Stokes type variational inequality problem on the fine mesh. Finally, numerical results are given to verify the theoretical analysis.     

Numerical analysis of a Picard multilevel stabilization of mixed finite volume method for the 2D/3D incompressible flow with large data

In this article, we develop a branch of nonsingular solutions of a Picard multilevel stabilization of mixed finite volume method for the 2D/3D stationary Navier‐Stokes equations without relying on the unique solution condition. The method presented consists of capturing almost all information of initial problem (the nonlinear problems) on the coarsest mesh and then performs one Picard defect correction (the linear problems) on each subsequent mesh based on previous information thus only solving one large linear systems. What is more, the method presented can results in a better coefficient matrix in the model presented with small viscosity. Theoretical results show that the method presented is derived with the convergence rate of the same order as the corresponding finite volume method/finite element method solving the stationary Navier‐Stokes equations on a fine mesh. Therefore, the method presented is definitely more efficient than the standard finite volume method/finite element method. Finally, numerical experiments clearly show the efficiency of the method presented for solving the stationary Navier‐Stokes equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 30–50, 2018     

Analysis of newton multilevel stabilized finite volume method for the three‐dimensional stationary Navier‐Stokes equations

This article proposes and analyzes a multilevel stabilized finite volume method(FVM) for the three‐dimensional stationary Navier–Stokes equations approximated by the lowest equal‐order finite element pairs. The method combines the new stabilized FVM with the multilevel discretization under the assumption of the uniqueness condition. The multilevel stabilized FVM consists of solving the nonlinear problem on the coarsest mesh and then performs one Newton correction step on each subsequent mesh thus only solving one large linear systems. The error analysis shows that the multilevel‐stabilized FVM provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the stationary Navier–Stokes equations on a fine mesh for an appropriate choice of mesh widths: h_j ～ h_j‐1², j = 1,…,J. Therefore, the multilevel stabilized FVM is more efficient than the standard one‐level‐stabilized FVM. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A stabilized multi-level method for non-singular finite volume solutions of the stationary 3D Navier–Stokes equations

This paper proposes and analyzes a stabilized multi-level finite volume method (FVM) for solving the stationary 3D Navier?CStokes equations by using the lowest equal-order finite element pair without relying on any solution uniqueness condition. This multi-level stabilized FVM consists of solving the nonlinear problem on the coarsest mesh and then performing one Newton correction step on each subsequent mesh, thus only solving a large linear system. An optimal convergence rate for the finite volume approximations of nonsingular solutions is first obtained with the same order as that for the usual finite element solution by using a relationship between the stabilized FVM and a stabilized finite element method. Then the multi-level finite volume approximate solution is shown to have a convergence rate of the same order as that of the stabilized finite volume solution of the stationary Navier?CStokes equations on a fine mesh with an appropriate choice of the mesh size: ${ h_{j} ~ h_{j-1}^{2}, j = 1,\ldots, J}$ . Finally, numerical results presented validate our theoretical findings.