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.     

SOME n-RECTANGLE NONCONFORMING ELEMENTS FOR FOURTH ORDER ELLIPTIC EQUATIONS

In this paper, three n-rectangle nonconforming elements are proposed with n ≥ 3. They are the extensions of well-known Morley element, Adini element and Bogner-Fox-Schmit element in two spatial dimensions to any higher dimensions respectively. These elements are all proved to be convergent for a model biharmonic equation in n dimensions.     

SOME ESTIMATES WITH NONCONFORMING ELEMENTS IN DOMAIN DECOMPOSITION ANALYSIS

1.IntroductionFOrsimplicityoftheexposition,weconsidertheellipticboundaryvalueproblemonaboundedopenpolygonaldomainfiCEZwhereItiswell--knownthat(1.1)hasauniquesolutionueH'(fl)(of.[7,15,16]).Supposethatfib~{e}isaquajsi--uniformmeshoffi,i.e.,fibsatisfieswhere…     

EXPLICIT ERROR ESTIMATES FOR MIXED AND NONCONFORMING FINITE ELEMENTS

In this paper, we study the explicit expressions of the constants in the error estimates of the lowest order mixed and nonconforming finite element methods. We start with an explicit relation between the error constant of the lowest order Raviart-Thomas interpolation error and the geometric characters of the triangle. This gives an explicit error constant of the lowest order mixed finite element method. Furthermore, similar results can be ex- tended to the nonconforming P1 scheme based on its close connection with the lowest order Raviart-Thomas method. Meanwhile, such explicit a priori error estimates can be used as computable error bounds, which are also consistent with the maximal angle condition for the optimal error estimates of mixed and nonconforming finite element methods.     

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 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.     

THE NONCONFORMING FINITE ELEMENT METHOD FOR SIGNORINI PROBLEM

We present the Crouzeix-Raviart linear nonconforming finite element approximation of the variational inequality resulting from Signorini problem. We show if the displacement field is of H2 regularity, then the convergence rate can be improved from O（h3/4） to quasi-optimal O（h｜log h｜1/4） with respect to the energy norm as that of the continuous linear finite element approximation. If stronger but reasonable regularity is available, the convergence rate can be improved to the optimal O（h） as expected by the linear approximation.     

A Locking-Free Scheme of Nonconforming Rectangular Finite Element for the Planar Elasticity

In this paper, the authors present a locking-free scheme of the lowest order nonconforming rectangle finite element method for the planar elasticity with the pure displacement boundary condition. Optimal order error estimate, uniformly for the Laméconstant $\lambda\in(0,\infty)$ is obtained.     

NON C~0 NONCONFORMING ELEMENTS FOR ELLIPTIC FOURTH ORDER SINGULAR PERTURBATION PROBLEM

In this paper we give a convergence theorem for non C^0 nonconforming finite element to solve the elliptic fourth order singular perturbation problem. Two such kind of elements, a nine parameter triangular element and a twelve parameter rectangular element both with double set parameters, are presented. The convergence and numerical results of the two elements are given.     

CONSTRAINED QUADRILATERAL NONCONFORMING ROTATED Q1 ELEMENT

In this paper, we define a new nonconforming quadrilateral finite element based on the nonconforming rotated Q1 element by enforcing a constraint on each element, which has only three degrees of freedom. We investigate the consistency, approximation, superclose property, discrete Green＇s function and superconvergence of this element. Moreover, we propose a new postprocessing technique and apply it to this element. It is proved that the postprocessed discrete solution is superconvergent under a mild assumption on the mesh.     

TRAPEZOIDAL PLATE BENDING ELEMENT WITH DOUBLE SET PARAMETERS

Using double set parameter method, a 12-parameter trapezoidal plate bending element is presented. The first set of degrees of freedom, which make the element convergent, are the values at the four vertices and the middle points of the four sides together with the mean values of the outer normal derivatives along four sides. The second set of degree of freedom, which make the number of unknowns in the resulting discrete system small and computation convenient are values and the first derivatives at the four vertices of the element. The convergence of the element is proved.     

ON LOCKING-FREE FINITE ELEMENT SCHEMES FOR THREE-DIMENSIONAL ELASTICITY

In the present paper,the authors discuss the locking phenomenon of the finite elementmethod for three-dimensional elasticity as the Laméconstant λ→∞.Three kinds of finiteelements are proposed and analyzed to approximate the three-dimensional elasticity withpure displacement boundary condition.Optimal order error estimates which are uniformwith respect to λ∈(0,+∞)are obtained for three schemes.Furthermore,numerical resultsare presented to show that,our schemes are locking-free and and the trilinear conformingfinite element scheme is locking.     

关于六角系统的极植问题

本文讨论六角系统中将宽度与直径结合起来的几个极值问题，并给出相应的极图构造。最后，指出并补正［４］中一个计数结果的不足。     

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.     

NONCONFORMING FINITE ELEMENT APPROXIMATIONS TOTHE UNILATERAL PROBLEM

1.IntroductionTherehavebeennumerousworkintheanalysisoffiniteelementmethodsfortheunilateralproblem(c.f.[4]andthereferencestherein).ItshouldbementionedthatinF.Scarpiniet.al.[6],I.Hlavaceket.al.[5]andF.Brezziet.al.[1],theconforminglinearelementapproximationtotheunilateralproblemhavebeenconsidered,andthevariouserrorboundshavebeenobtainedinthedifferentassumptionofregularityofsolutionoftheproblem.Inthispaper3weconsiderthreenonconformingfiniteelement(i.e.twoCrouzrixRaviartlinearelementsandWilsone…     

基于插值方法构造对称的矩形板单元

1　引　　言在有限元方法中,构造多项式类有限元的问题可以归结为多元多项式插值问题.在多元多项式插值的情形中,插值条件与插值多项式空间之间存在所谓的“匹配”(correct)问题,参见deBoor的论文[9].在构造非协调有限元时,关键的是设计适当的有限元形参数和适当的有限元形函数空间,再设计与之匹配的形参数.在经典的有限元构造方法中,例如,基于位移假设的板元的构造,就是首先定义好形函数空间.在这种情况下,所设计形参数必须满足两个条件:(1)形参数作为插值条件,必须与事先给定的形函数空间是插值匹配的,也即,由形参数定义的插值条件在形函…     

完全三次非协调板元的误差估计

本文考虑以下重调和方程的边值问题:△~2u=f,在G上,u=?u/?n=0,在?G上,其中G为R~2上多边形区域,n为单位外法向量.此问题的变分形式为:找u∈H_0~2(G),使得: α(u,v)=(f,vv)?v∈ _0~~2(G),其中 α(u,v)=∫_G[△u△v+(1-σ)(2u_(x_1x_2)v_(x_1x_2)-u(x_1x_1)v_(x_2x_2)-u_(x_2x_2)v_(x_1x_1))]dx_1dx_2 设τ_h为G的一致正则矩形剖分,h为所有元的最大直径.文[5]中构造了一个完全三次非协调板元,它的形函数为完全三次多项式;自由度集合由函数在四个顶点之值及两个三阶     

UNIFORMLY CONVERGENT NONCONFORMING ELEMENT FOR 3-D FOURTH ORDER ELLIPTIC SINGULAR PERTURBATION PROBLEM

In this paper, using a bubble function, we construct a cuboid element to solve the fourth order elliptic singular perturbation problem in three dimensions. We prove that the nonconforming CO-cuboid element converges in the energy norm uniformly with respect to the perturbation parameter.     

ON THE SELECTION OF SHAPE FUNCTION SPACES OF TRIANGULAR PLATE ELEMENTS

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