一类非协调元的收敛性分析

１．引言最近,［１］在求解Ｓｔｏｋｅｓ方程时提出了一类非协调元．这类非协调元定义在矩形网格上,形式简单,自由度少,比熟知的多线性元还少．例如,对于三维问题,它们只有六个自由度,是型函数在各个表面中点处的函数值或者表面上的平均值．与一些熟知的非协调元不同之处是,由于这类非协调元简单的形式及其非常少的自由度,它们不包含任何协调的部分,但是,已有的结果表明它们具有良好的数值表现［１］,［２］中已有一些理论分析,用它们来求解晶体的微结构方程,也取得较好的结果．本文将详细分析这类非协调元的收敛性,所用的工…     

二阶椭圆问题的稳定化杂交有限元分析

0　引　　言Raviart&Thomas(1977)[13]基于Babǔska-Brezzi有限元理论[1][5]发展了二阶椭圆问题的基本杂交方法.该文指出,为确定合适的自由度,一般将杂交元刻划为非协调元.然而,对三角形偶数次杂交元和四边形杂交元而言,[13]是通过扩充手段克服有限维空间“匹配”问题的.由于扩充元的复杂性及其不再能刻划为非协调元,以致于实际计算无法选取自由度.Thomas的博士论文[15]提供了一个解决办法.即利用Gauss-Legendre数值求积分公式将扩充元近似刻划成非协调元,得到数值积分意义下的杂交方法.如此处理虽然大大简化了原杂交格式的求解过程,但数…     

四维空间上的C~1单纯形有限元族

我们基于四维空间一般单纯形网格构造了一族可微的分片k(k=17)次多项式有限元.这类光滑有限元空间具有最优阶逼近性.作为副产品,我们得到一族三维的四面体网格上C~2-Pk有限元.     

SURFACE FINITE ELEMENTS FOR

In this article we define a surface finite element method （SFEM） for the numerical solution of parabolic partial differential equations on hypersurfaces F in R^n＋1. The key idea is based on the approximation of F by a polyhedral surface Гh consisting of a union of simplices （triangles for n = 2, intervals for n = 1） with vertices on Г. A finite element space of functions is then defined by taking the continuous functions on Гh which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on Г. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. Numerical experiments are described for several linear and nonlinear partial differential equations. In particular the power of the method is demorrstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow.     

A SUPERCONVERGENT ESTIMATE OF WILSON-LIKE ELEMENTS

In this paper, the supercouvergence of a class of Wilson-like elements is considered, and a superconvergent estimate of Wilson-like elements is obtained for strong regular meshes.     

具有很多素数阶元的有限群

考察元素的阶如何影响有限群的结构是群论中的一个重要课题．本文研究存在一个正规子群N，N外的元素都是素数阶元的有阶群．主要利用熟知的Thompson的一个定理，获得了这样的有限群．     

粘性不可压缩对流占优Oseen方程的非协调子格稳定化有限元法分析

<正>1引言本文考虑粘性不可压缩对流占优Oseen方程,(?)(1)其中:Ω(?)R~d(d=2)为具有Lipschitz连续边界的有界开集,β∈W~(1,∞)(Ω)且▽·β=0,μ、σ为常数,f∈L~2(Ω).当采用通常的混合有限元方法(MFEM)求解时,一般会遇到以下两个困难:·为保证速度和压力数值解稳定,要求有限元空间满足inf-sup(or Babuska-Brezzi)条件.·当对流占优,即0μ《||β||_(L~∞(Ω))时,数值解会产生伪振荡.     

AN ERROR ANALYSIS METHOD SPP-BEAM AND A CONSTRUCTION GUIDELINE OF NONCONFORMING FINITE ELEMENTS FOR FOURTH ORDER ELLIPTIC PROBLEMS

Under two hypotheses of nonconforming finite elements of fourth order elliptic problems,we present a side-patchwise projection based error analysis method(SPP-BEAM for short).Such a method is able to avoid both the regularity condition of exact solutions in the classical error analysis method and the complicated bubble function technique in the recent medius error analysis method.In addition,it is universal enough to admit generalizations.Then,we propose a sufficient condition for these hypotheses by imposing a set of in some sense necessary degrees of freedom of the shape function spaces.As an application,we use the theory to design a P3 second order triangular H2 non-conforming element by enriching two P4 bubble functions and,another P4 second order triangular H2 nonconforming finite element,and a P3 second order tetrahedral H2 non-conforming element by enriching eight P4 bubble functions,adding some more degrees of freedom.     

A CLASS OF NEW NONCONFORMING FINITE ELEMENTS

A class of new nine-parameter nonconforming finite elements with convergent prop-erties are constructed in terms of the approach of interpolation methods in this paper. All of those finite elements can be viewed as the generalization of Zienkiewicz's finite element by dis-lurbing its parameters.     

抛物型方程一个新的非协调混合元超收敛性分析及外推

本文研究了抛物型方程在新混合元格式下的非协调混合有限元方法. 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,运用高精度分析和对时间t的导数转移技巧,借助于插值后处理技术,分别导出了关于原始变量u的H¹-模和通量p=▽u在L²-模下的O（h²）阶超逼近性质和整体超收敛. 进一步,通过构造合适的辅助问题,运用Richardson 外推格式,得到了具有更高精度O（h³）阶的外推结果. 最后,给出了一些数值结果验证了理论分析的正确性.     

NATURAL SUPERCONVERGENT POINTS OF EQUILATERAL TRIANGULAR FINITE ELEMENTS - A NUMERICAL EXAMPLE

A numerical test case demonstrates that the Lobatto and the Gauss points are notnatural superconvergent points of the cubic and the quartic finite elements under equilateraltriangular mesh for the Poisson equation.     

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.     

两类各向异性非协调元的某些超收敛性质分析

在各向异性网格下,讨论了两类非协调矩形元对二阶椭圆边值问题的某些超逼近性和超收敛性,并证明了在单元中心点这种超收敛性仅为一种点态现象.数值结果验证了我们理论分析的正确性.     

常微分方程初值问题连续有限元的超收敛性

1 引言及算法 考虑一阶非线性常微分方程初值问题u′=f(t,u),t∈I=[0,T],u(0)=u_0,(1)其中f(t,u)是t,u的适当光滑函数。我们知道,常微分方程初值问题的数值解法不仅本身有独立的兴趣,它也是抛物与双曲方程时间离散的基础。目前已有许多数值方法,如     

A NEW NONCONFORMING MIXED FINITE ELEMENT SCHEME FOR THE STATIONARY NAVIER-STOKES EQUATIONS

In this article, a new stable nonconforming mixed finite element scheme is proposed for the stationary Navier-Stokes equations, in which a new low order CrouzeixRaviart type nonconforming rectangular element is taken for approximating space for the velocity and the piecewise constant element for the pressure. The optimal order error estimates for the approximation of both the velocity and the pressure in L2-norm are established, as well as one in broken H1-norm for the velocity. Numerical experiments are given which are consistent with our theoretical analysis.     

ON THE SELECTION OF SHAPE FUNCTION SPACES OF TRIANGULAR PLATE ELEMENTS

ＯＮＴＨＥＳＥＬＥＣＴＩＯＮＯＦＳＨＡＰＥＦＵＮＣＴＩＯＮＳＰＡＣＥＳＯＦＴＲＩＡＮＧＵＬＡＲＰＬＡＴＥＥＬＥＭＥＮＴＳＣｈｅｎＳｈａｏｃｈｕｎ；ＳｈｉＤｏｎｇｙａｎｇ（Ｄｅｐｔ．ｏｆＳｙｓ．Ｓｃｉ．＆Ｍａｔｈ．，ＺｈｅｎｇｚｈｏｕＵｎｉｖ．，Ｚｈｅ...     

关于几种三角形板元等价性的注记

1．引言求解薄板弯曲问题的广义协调元法近年来有了发展,本文讨论几种三角形板元之间的等价关系．我们证明,[2]中构造的GCⅢ－T9元等价于Specht元问．同一作者在[4]中构造的GCⅡ－T9元实际上也等价于Specht元,因此,它们彼此等价．另外,[7]中用对称列式构造的一个广义协调元是和作者在问中所引进的单元等价．例已指出,[7]中的那个元等价于问中的VZ1元,所以[7]和[6]中的两个广义协调元均等价于已知的VZ1元．2．广义协调GCⅢ－T9单元设三角形单元K的三个顶点依反时针方向分别为p i（xi,yi）,节点参数是pi点处的函数值及其一阶导…     

A POSTERIORI ERROR ESTIMATES IN ADINI FINITE ELEMENT FOR EIGENVALUE PROBLEMS

In this paper, we discuss a posteriori error estimates of the eigenvalue $\lambda_h$ given by Adini nonconforming finite element. We give an asymptotically exact error estimator of the $\lambda_h$. We prove that the order of convergence of the $\lambda_h$ is just 2 and the $\lambda_h$ converge from below for sufficiently small $h$.     

HIGH ACCURACY FINITE VOLUME ELEMENT METHOD FOR TWO-POINT BOUNDARY VALUE PROBLEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

1　ＩｎｔｒｏｄｕｃｔｉｏｎＦｉｎｉｔｅｖｏｌｕｍｅｅｌｅｍｅｎｔｍｅｔｈｏｄｏｒＦＶＥＭｕｓｅｓａｖｏｌｕｍｅｉｎｔｅｇｒａｌｆｏｒｍｕｌａｔｉｏｎｏｆｔｈｅｄｉｆ ｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｗｉｔｈａｆｉｎｉｔｅｐａｒｔｉｔｉｏｎｉｎｇｓｅｔｏｆｖｏｌｕｍｅｔｏｄｉｓｃｒｅｔｉｚｅｔｈｅｅｑｕａｔｉｏｎ ,ｔｈｅｎｒｅ ｓｔｒｉｃｔｓｔｈｅａｄｍｉｓｓｉｂｌｅｆｕｎｃｔｉｏｎｓｔｏａｌｉｎｅａｒｆｉｎｉｔｅｅｌｅｍｅｎｔｓｐａｃｅｔｏｄｉｓｃｒｅｔｉｚｅｔｈｅｓｏｌｕｔｉｏｎ ([1 -4 ] ) .…     

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