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…     

SOME SUPERCONVERGENCE RESULTS OF WILSON-LIKE ELEMENTS

1.IntroductionThesuperconvergenceoffiniteelementmethodsisanimportantpropertybothintheoryandinpractice.Manysuperconvergenceresultsaboutconformingfiniteelementmethodshavebeenobtained(see[4][17]).Buttherearealsomanynonconformingfiniteelementmethodsincomputat…     

A CANONICAL CONSTRUCTION OF $H^m$-NONCONFORMING TRIANGULAR FINITE ELEMENTS

We design a family of 2D H^m-nonconforming finite elements using the full P_2m-3 degree polynomial space, for solving $2m$th elliptic partial differential equations. The consistent error is estimated and the optimal order of convergence is proved. Numerical tests on the new elements for solving tri-harmonic, 4-harmonic, 5-harmonic and 6-harmonic equations are presented, to verify the theory.     

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.     

Anisotropic Superconvergence Analysis for the Wilson Nonconforming Element

1 Introduction The Wilson nonconforming element has been widely used in computational mechanics and struc- tural engineering because of its good convergence. In many practical cases, it seems better than the bilinear conforming finite element. This phenomenon causes the great interest of many people who study finite elements. Some papers about the Wilson element have been published which deal with superconvergence. In [6], the superclose property and the global superconvergence are obtained …     

MULTIGRID FOR THE MORTAR ELEMENT METHOD WITH LOCALLY P_1 NONCONFORMING ELEMENTS

In this paper we study the theoretical properties of multigrid algorithm for discretization of the Poisson equation in 2D using a mortar element method under the assumption that the triangulations on every subdomain are uniform. We prove the convergence of the W-cycle with a sufficiently large number of smoothing steps. The     

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.     

A local parallel superconvergence method for the incompressible flow by coarsening projection

In this article, we investigate a local parallel superconvergence method by coarsening projection for the incompressible Stokes flow. The method is a combination of the local superconvergence technique and the given framework of local parallel method. For the smooth subdomains, the local superconvergence method is applied in a higher order finite dimensional space corresponding to an appropriate coarse mesh on interior domain. Moreover, a useful and flexible local parallel method is designed to obtain the local parallel superconvergence results of presented method, which offset theoretical limitation of the model without the smoothness of the exact solution and a priori regularity of the underlying problem over the whole domain. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1209–1223, 2015     

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     

APPLICATION OF SUPERCONVERGENCE TO A MODEL FOR COMPRESSIBLE MISCIBLE DISPLACEMENT

A modification of a finite element method of Douglas and Roberts for approximating the solution of the equations describing compressible miscible displacement in a porous medium is proposed and analyzed. The pressure is treated by a parabolic mixed finite element method using a Raviart-Thomas space of index rover a quasiregular partition, An extension of the Darcy velocity along Gauss lines is used in the evaluation of the coefficients in the Galerkin procedure for the concentration. A simple computational procedure allows the superconvergence property of the fluid velocity to be retained in our total algorithm.     

A coupling of nonconforming and mixed finite element methods for Biot's consolidation model

In this article, we develop a nonconforming mixed finite element method to solve Biot's consolidation model. In particular, this work has been motivated to overcome nonphysical oscillations in the pressure variable, which is known as locking in poroelasticity. The method is based on a coupling of a nonconforming finite element method for the displacement of the solid phase with a standard mixed finite element method for the pressure and velocity of the fluid phase. The discrete Korn's inequality has been achieved by adding a jump term to the discrete variational formulation. We prove a rigorous proof of a‐priori error estimates for both semidiscrete and fully‐discrete schemes. Optimal error estimates have been derived. In particular, optimality in the pressure, measured in different norms, has been proved for both cases when the constrained specific storage coefficient c₀ is strictly positive and when c₀ is nonnegative. Numerical results illustrate the accuracy of the method and also show the effectiveness of the method to overcome the nonphysical pressure oscillations. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

EDGE-ORIENTED HEXAGONAL ELEMENTS

In this paper, two new nonconforming hexagonal elements are presented, which are based on the trilinear function space Q1^（3） and are edge-oriented, analogical to the case of the rotated Q1 quadrilateral element. A priori error estimates are given to show that the new elements achieve first-order accuracy in the energy norm and second-order accuracy in the L^2 norm. This theoretical result is confirmed by the numerical tests.     

A review of two different approaches for superconvergence analysis

In 1995, Wahbin presented a method for superconvergence analysis called Interior symmetric method, and declared that it is universal. In this paper, we carefully examine two superconvergence techniques used by mathematicians both in China and in America. We conclude that they are essentially different.     

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.     

Superconvergence analysis of splitting positive definite nonconforming mixed finite element method for pseudo-hyperbolic equations

In this paper, a new splitting positive definite nonconforming mixed finite element method is proposed for pseudo-hyperbolic equations, in which a quasi-Wilson quadrilateral element is used for the flux p, and the bilinear element is used for u. Superconvergence results in ｜｜·｜｜div,h norm for p and optimal error estimates in L2 norm for u are derived for both semi-discrete and fully discrete schemes under almost uniform meshes.     

A superconvergent nonconforming mixed finite element method for the Navier–Stokes equations

The superconvergence for a nonconforming mixed finite element approximation of the Navier–Stokes equations is analyzed in this article. The velocity field is approximated by the constrained nonconforming rotated Q₁ (CNRQ₁) element, and the pressure is approximated by the piecewise constant functions. Under some regularity assumptions, the superconvergence estimates for both the velocity in broken H¹‐norm and the pressure in L²‐norm are obtained. Some numerical examples are presented to demonstrate our theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 646–660, 2016     

Sobolev方程的一类各向异性非协调有限元逼近

在各向异性网格下,分别讨论了Sobolev方程在半离散和全离散格式下的一类非协调有限元逼近,得到了与传统有限元方法相同的误差估计和一些超逼近性质.同时在半离散格式下,通过构造具有各向异性特征的插值后处理算子得到了整体超收敛结果.     

p—型有限元方法的L^2模误差估计的一个补充

本文讨论Kenneth Erikssion提出的模型问题的p-型有限元方法,解决了文[1]定理2后提出的问题,并给出提高误差收敛阶的一个方法。     

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…