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.     

定常Navier-Stokes方程的一个二阶非协调三角型混合元格式（英文）

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 H~1-norm and L~2-norm for velocity as well as the L~2-norm for the pressure are derived.     

Nonconforming mixed finite element approximation to the stationary Navier-Stokes equations on anisotropic meshes

The main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary Navier-Stokes equations under anisotropic meshes. That is, the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure. The convergence analysis is presented and the optimal error estimates both in a broken H¹-norm for the velocity and in an L²-norm for the pressure are derived on anisotropic meshes.     

非定常Navier-Stokes方程的质量集中各向异性非协调有限元逼近

该文将一个低阶Crouzeix-Raviart型非协调三角形元应用到非定常Navier-Stokes方程,给出了其质量集中有限元逼近格式.在不需要传统Ritz-Volterra投影下,通过引入两个辅助有限元空间对边界进行估计的技巧,在各向异性网格下导出了速度的L~2模和能量模及压力的L~2模的误差估计.     

SUPERCONVERGENCE ANALYSIS OF LOW ORDER NONCONFORMING MIXED FINITE ELEMENT METHODS FOR TIME-DEPENDENT NAVIER-STOKES EQUATIONS

In this paper,the superconvergence properties of the time-dependent Navier-Stokes equations are investigated by a low order nonconforming mixed finite element method(MFEM).In terms of the integral identity technique,the superclose error estimates for both the velocity in broken H-norm and the pressure in L2-norm are first obtained,which play a key role to bound the numerical solution in Lx-norm.Then the corresponding global superconvergence results are derived through a suitable interpolation postprocessing approach.Finally,some numerical results are provided to demonstrated the theoretical analysis.     

The Crouzeix-Raviart type nonconforming finite element method for the nonstationary Navier-Stokes equations on anisotropic meshes

This paper is devoted to study the Crouzeix-Raviart （C-R） type nonconforming linear triangular finite element method （FEM） for the nonstationary Navier-Stokes equations on anisotropic meshes. By intro- ducing auxiliary finite element spaces, the error estimates for the velocity in the L2-norm and energy norm, as well as for the pressure in the L2-norm are derived.     

CONVERGENCE ANALYSIS FOR A NONCONFORMING MEMBRANE ELEMENT ON ANISOTROPIC MESHES

Regular assumption of finite element meshes is a basic condition of most analysis offinite element approximations both for conventional conforming elements and nonconform-ing elements.The aim of this paper is to present a novel approach of dealing with theapproximation of a four-degree nonconforming finite element for the second order ellipticproblems on the anisotropic meshes.The optimal error estimates of energy norm and L~2-norm without the regular assumption or quasi-uniform assumption are obtained based onsome new special features of this element discovered herein.Numerical results are givento demonstrate validity of our theoretical analysis.     

非线性抛物方程的质量集中非协调元分析

将一个低阶Crouzeix-Raviart型非协调三角形元应用到一类非线性抛物方程,并建立了质量集中的半离散和向后Euler全离散逼近格式,在一般各向异性网格上利用插值算子导出了L²-模的最优误差估计.     

热传导方程一个新的H~1-Galerkin非协调混合有限元格式

对热传导方程提出了一个新的H~1-Galerkin非协调混合有限元格式,其逼近空间不需满足LBB相容性条件,且在不引进传统的Rutz投影的情况下,得到了与以往协调有限元方法相同的L~2-模和H~1-模的误差估计.     

On the relationship between finite volume and finite element methods applied to the Stokes equations

We investigate the relationship between finite volume and finite element approximations for the lower‐order elements, both conforming and nonconforming for the Stokes equations. These elements include conforming, linear velocity‐constant pressure on triangles, conforming bilinear velocity‐constant pressure on rectangles and their macro‐element versions, and nonconforming linear velocity‐constant pressure on triangles and nonconforming rotated bilinear velocity‐constant pressure on rectangles. By applying the relationship between the two methods, we obtain the convergence finite volume solutions for the Stokes equations. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 440–453, 2001.     

Convergence and stability of a stabilized finite volume method for the stationary Navier-Stokes equations

In this paper, a new stabilized finite volume method is studied and developed for the stationary Navier-Stokes equations. This method is based on a local Gauss integration technique and uses the lowest equal order finite element pair P ₁–P ₁ (linear functions). Stability and convergence of the optimal order in the H ¹-norm for velocity and the L ²-norm for pressure are obtained. A new duality for the Navier-Stokes equations is introduced to establish the convergence of the optimal order in the L ²-norm for velocity. Moreover, superconvergence between the conforming mixed finite element solution and the finite volume solution using the same finite element pair is derived. Numerical results are shown to support the developed convergence theory.     

A class of nonconforming quadrilateral finite elements for incompressible flow

This paper focuses on the low-order nonconforming rectangular and quadrilateral finite elements approximation of incompressible flow.Beyond the previous research works,we propose a general strategy to construct the basis functions.Under several specific constraints,the optimal error estimates are obtained,i.e.,the first order accuracy of the velocities in H1-norm and the pressure in L2-norm,as well as the second order accuracy of the velocities in L2-norm.Besides,we clarify the differences between rectangular and quadrilateral finite element approximation.In addition,we give several examples to verify the validity of our error estimates.     

EFK方程一个新的低阶非协调混合有限元方法的高精度分析

对Extended Fisher-Kolmogorov(EFK)方程,利用EQ_1~(rot)元和零阶RaviartThomas(R-T)元建立了一个新的非协调混合元逼近格式.首先,证明了半离散格式逼近解的一个先验估计并证明了逼近解的存在唯一性.在半离散格式下,利用上述两种元的高精度分析结果以及这个先验估计,在不需要有限元解u_h属于L~∞的条件下,得到了原始变量u和中间变量v=-?u的H~1-模以及流量p=u的(L~2)~2-模意义下O(h~2)阶的超逼近性质.同时,借助插值后处理技术,证明了上述变量的具有O(h~2)阶的整体超收敛结果.其次,建立了一个新的线性化向后Euler全离散格式并证明了其逼近解的存在唯一性.另一方面,通过对相容误差和非线性项采取与传统误差分析不同的新的分裂技巧,分别导出了以往文献中尚未涉及的关于u和v在H~1-模以及p在(L~2)~2-模意义下具有O(h~2+τ)阶的超逼近性质,进一步地,借助插值后处理技术,得到了上述变量的整体超收敛结果.这里h和τ分别表示空间剖分参数和时间步长.最后,给出了一个数值算例,计算结果验证了理论分析的正确性.     

A locking-free anisotropic nonconforming rectangular finite element approximation for the planar elasticity problem

This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2-norm are obtained. The restrictions of regularity assumption and quasi-uniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended.     

Stokes问题的一个非协调有限元

<正>1引言定常Stokes问题是流体力学中的一种重要问题,是标准的混合问题,速度与压力同时计算.关于该问题有限元求解的文章很多([1],[2],[4],[5],[6],[8],[9]),分析的难点在于单元必须满足离散的Babuska-Brezzi([2])条件.在([4])中提出了著名的非协调Crouzeix-Raviart     

Simple nonconforming brick element for 3D Stokes equations

A simple nonconforming brick element is proposed for 3D Stokes equations. This element has 15 degrees of freedom and reaches the lowest approximation order. In the mixed scheme for Stokes equations, we adopt our new element to approximate the velocity, along with the discontinuous piecewise constant element for the pressure. The stability of this scheme is proved and thus the optimal convergence rate is achieved. A numerical example verifies our theoretical analysis.     

纯应力平面弹性问题的一个非协调矩形元(英文)

针对纯应力平面弹性问题构造了一个非协调矩形元.该单元满足离散的第二Korn不等式,并且关于λ有一致最优收敛阶,其误差的能量模和L2-模分别为O(h2)和O(h3) .     

二阶椭圆问题的各向异性非协调Crouzeix-Raviart型有限元方法

对满足最大角条件和坐标系条件的二维区域中的各向异性一般三角形网格,研究了二阶椭圆问题的非协调Crouzeix-Raviart型线性三角形有限元逼近,得到了最优的能量模和L2-模误差估计结果．     

Stokes问题的杂交-混合有限元分析

本文发展Stokes问题的一个四变量杂交-混合变分方程：应力-速度-压力-拉格朗日乘子．然后发展其有限元方法：对应四变量分别用间断型Raviart—Thomas最低阶元，分片常数元，连续线性元和连续线性元的迹空间．我们获得了稳定性和最优误差界．通过后处理办法，我们得到一个适合于计算的速度-压力格式，该格式可视为“Mini”元方法的一个变形(本文格式中引入了局部投影算子)．然而，本文格式关于压力具有“超收敛”结果：得到了压力关于H^1-范的误差界O(h)．     

对流扩散方程的稳定化流线扩散法最小二乘非协调有限元分析

将最小二乘法和稳定化的流线扩散法相结合,研究了对流扩散方程的非协调有限元格式,用矩形EQ_1~(rot)元和零阶R-T元分别来逼近位移和应力,利用单元本身的特殊性质,证明了离散格式解的存在惟一性,得到了位移H~1-模和应力H(div)-模的最优误差估计.