Stokes问题在各向异性网格下的Bernadi-Raugel有限元逼近

在各向异性网格下,得到了Stokes问题著名的Bernadi-Raugel混合有限元格式的超逼近性质,而且通过构造插值后处理算子得到了关于速度的超收敛结果.     

Stokes问题的一种新的混合有限元逼近

为了求解Ｓｔｏｋｅｓ问题，本文构造出了一类新的满足ＢＢ－条件的有限元空间对，并给出了相应的超收敛分析．     

Stokes问题有限元逼近中求导数的提取法

1.引言 我们知道Poisson方程和平面弹性问题的解的导数的近似值可以通过所谓提取公式得到,而不必对近似解直接求导数.这样我们可以得到具有与近似解本身同阶精度的导数的近似值.这一方法已被用于基于插值误差的后验误差估计及相应的自适应有限元方法中本文将这一方法应用于Stokes问题的有限元逼近,从Stokes方程的解的     

Stokes问题的各向异性误差分析

对二维空间Stokes问题提出了各向异性平行四边形混合有限元逼近格式,证明了其在不要求满足正则性和拟一致条件下的收敛性以及在各向异性条件下对相容误差部分的超收敛估计.     

问题变分不等式的一类各向异性Crouzeix-Raviartart型有限元逼近

本文研究了Signorini变分不等式问题的一类各向异性Crouzeix-Raviart型非协调有限元逼近。通过一些新的技巧，得到了相应的最优误差估计。     

Stokes问题各向异性网格Q2-P1混合元超收敛分析

讨论Stokes问题在各向异性冈格下的Q2-P1混合有限元方法，利用积分恒等式技巧得到了与传统方法相同的超逼近性质，同时基于插值后处理的技巧，构造了速度和压力的一对插值后处理算子，并且前者具有备向异性特征，从而导出了整体超收敛结果．     

Plate Contact问题的混合有限元逼近

论文考虑了Plate Contact问题的混合有限元逼近,其变分问题为第二类四阶椭圆变分不等问题.首先,根据正则化方法,得到原问题的正则化问题.再根据网格依赖范数技巧,考虑了正则化问题的Ciarlet-Raviart混合有限元逼近,并证明了真解与逼近解之间的误差估计.最后通过数值算例验证了理论分析的结果.     

Stokes问题非协调混合有限元超收敛分析

本文通过引入全新的技巧,研究了Stokes问题的非协调混合有限元方法,得到了关于速度与压力的超逼近性质.进一步地通过构造一个恰当的插值后处理算子,还得到了关于速度的整体超收敛结果.     

Stokes问题非协调有限元逼近的最大模估计

本文考察了二维稳态和非稳态Stokes问题的基于速度—压力形式的非协调C-R逼近格式,利用Sobolev权模技巧和权模LBB条件,得到了稳态问题速度(包括它的梯度)和压力逼近解的拟最优的最大模估计,利用稳态问题结果和Stokes投影技巧,得到了非稳态问题速度(包括它的梯度)和压力的半离散逼近解的拟最优的最大模估计。     

CONVERGENCE OF A MIXED FINITE ELEMENT FOR THE STOKES PROBLEM ON ANISOTROPIC MESHES

The main aim of this paper is to study the convergence properties of a low order mixed finite element for the Stokes problem under anisotropic meshes. We discuss the anisotropic convergence and superconvergence independent of the aspect ratio. Without the shape regularity assumption and inverse assumption on the meshes, the optimal error estimates and natural superconvergence at central points are obtained. The global superconvergence for the gradient of the velocity and the pressure is derived with the aid of a suitable postprocessing method. Furthermore, we develop a simple method to obtain the superclose properties which improves the results of the previous works .     

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.     

C~1连续的双3次双4次Hermite元各向异性分析

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各向异性网格下抛物方程一个新的非协调混合元收敛性分析

本文将 Crouzeix-Raviart 型非协调线性三角形元应用到抛物方程,建立了一个新的混合元格式.在抛弃传统有限元分析的必要工具 Ritz 投影算子的前提下,直接利用单元的插值性质和导数转移技巧, 分别得到了各向异性剖分下关于原始变量u 的H^-1-模和积分意义下L²-模以及通量p=-▽u 在L²-模下的最优阶误差估计.数值结果与我们的理论分析是相吻合的.     

CONVERGENCE ANALYSIS OF MORLEY ELEMENT ON ANISOTROPIC MESHES

The main aim of this paper is to study tile convergence of a nonconforming triangular plate element-Morley element under anisotropic meshes. By a novel approach, an explicit bound for the interpolation error is derived for arbitrary triangular meshes （which even need not satisfy the maximal angle condition and the coordinate system condition ）, the optimal consistency error is obtained for a family of anisotropically graded finite element meshes.     

A POSTERIORI ERROR ESTIMATION OF THE NEW MIXED ELEMENT SCHEMES FOR SECOND ORDER ELLIPTIC PROBLEM ON ANISOTROPIC MESHES

This paper presents a posteriori residual error estimator for the new mixed el-ement scheme for second order elliptic problem on anisotropic meshes. The reliability and efficiency of our estimator are established without any regularity assumption on the mesh.     

各向异性网格下Bogner-Fox-Schmit元的超收敛

1引言 有限元超收敛的研究已有三十多年的历史,至今为止已取得了丰富的成果,可见[3],[18],[10],[6],[5]以及[17].1981年,陈传淼(见[2]345-372页)考虑了四阶板问题有限元解的超收敛性,得到了高一阶的超收敛结果.1995年,林群和罗平[8]用积分恒等式技巧再次研究这个问题,在均匀矩形网格的条件下,得到了更好的结论,有限元解与有限元插值函数之间的误差在H2范数下,有高二阶的超收敛.     

ERROR ANALYSIS OF A STABILIZED FINITE ELEMENT METHOD FOR THE GENERALIZED STOKES PROBLEM

This paper is devoted to the establishment of sharper $a$ $priori$stability and error estimates of a stabilized finite element method proposed by Barrenechea and Valentin for solving the generalized Stokes problem, which involves a viscosity $\nu$ and a reaction constant $\sigma$. With the establishment of sharper stability estimates and the help of $ad$ $hoc$finite element projections, we can explicitly establish the dependence of error bounds of velocity and pressure on the viscosity $\nu$, the reaction constant $\sigma$, and the mesh size $h$. Our analysis reveals that the viscosity $\nu$ and the reaction constant $\sigma$ respectively act in the numerator position and the denominator position in the error estimates of velocity and pressure in standard norms without any weights. Consequently, the stabilization method is indeed suitable for the generalized Stokes problem with a small viscosity $\nu$ and a large reaction constant $\sigma$. The sharper error estimates agree very well with the numerical results.     

ON THE P1 POWELL-SABIN DIVERGENCE-FREE FINITE ELEMENT FOR THE STOKES EQUATIONS

The stability of the P1-P0 mixed-element is established on general Powell-Sabin triangular grids. The piecewise linear finite element solution approximating the velocity is divergence-free pointwise for the Stokes equations. The finite element solution approximating the pressure in the Stokes equations can be obtained as a byproduct if an iterative method is adopted for solving the discrete linear system of equations. Numerical tests are presented confirming the theory on the stability and the optimal order of convergence for the P1 Powell-Sabin divergence-free finite element method.