ON THE FINITE VOLUME ELEMENT VERSION OF RITZ-VOLTERRA PROJECTION AND APPLICATIONS TO RELATED EQUATIONS

In this paper, we present a general error analysis framework for the finite volume element (FVE) approximation to the Ritz-Volterra projection, the Sobolev equations and parabolic integro-differential equations. The main idea in our paper is to consider the FVE methods as perturbations of standard finite element methods which enables us to derive the optimal L2 and H1 norm error estimates, and the L∞ and W∞1 norm error estimates by means of the time dependent Green functions. Our disc ussions also include elliptic and parabolic problems as the special cases.     

SUPERCONVERGENCE ANALYSIS FOR CUBIC TRIANGULAR ELEMENT OF THE FINITE ELEMENT SUPERCONVERGENCE ANALYSIS FOR CUBIC TRIANGULAR ELEMENT OF THE FINITE ELEMENT

1. IntroductionAlthough we bed proved the superconvergence of quadratic triangular elem6ntsbefore 1985, the superconvergence reseaxch of k(k 2 3)-degree triangulax elemeds Onlyhas a few advances, e.g.,.Lin, Yan and Zhou (see [15]) Prove that the three degreeHerlnite elements possess superconvergence and Walilbin (see [5-7]) obtains a roughresult by using a fine interinr estimation, that is, the placement fUnCtion or its gradestmay have weak superconvergence in the lOcal syUUnetric points,e…     

Webster方程有限元法的超收敛性

<正>1前言由语音生成过程建立的数学模型是一个无界区域上的声压波动方程,引入人工边界可将这个无界区域分成一个内部有界区域和另一个无界区域,再将时间和空间进行变量分离,就可以得到一个如下一维简单的基于内部有界区间的微分方程边值问题     

一类波动方程有限元投影格式的误差分析

1.引 言 本文考虑如下不含阻尼项的波动方程的有限元逼近: 其中区域Ω Rd（d=2,3）是足够光滑的有界多边形区域,其边界为Γ=　Ω.初始条件为:当t=0时,u=u0,ut=u1.     

THE STABILITY AND APPROXIMATION PROPERTIES OF RITZ-VOLTERRA PROJECTION AND APPLICATION,I

The purpose of this paper is to study the stability and approximation properties of Ritz-Volterra projection. Through constructing a new type of Green functions and making use of various properties and estimates related with the functions, we prove that the Ritz-Volterra projection defined on the finite-dimensional subspace S_h of H_o~1 possesses the W_p~1-stability and the optimal approxi mation properties in W_p~1 and L_p for 2≤p≤∞. Our results, in this paper, can be applied to the finite element approximations for many evolution equations such as parabolic and hyperbolic integrodifferential equations,Sobolevequations and visco-elasticity, etc.     

GLOBAL SUPERCONVERGENCE ESTIMATES OF FINITE ELEMENT METHOD FOR SCHRODINGER EQUATION

1.IntroductionConsiderthefollowinginitialboundaryvalueproblemofSchr6dingerequationwheren~[0,1]',at~On/Ot,T>0isaconstant.Theequivalentvariationalformof(1.1)is:foralltE[0,T],findu(t)6Hi(n)satisfiesthefollowingvariationalequation:where(w,v)~IwvdxdenotestheinnerproductofL'(fl)anda(u,v)~(Vu,Vv),ibetheimaginaryunit.Weassumethatthefunctionsarecomplex--valuedandHibertspacesarecomplexspaces.LetThbeaquasiuniformrectangulationoffiwithmeshsizeh>0andS'(O)CHi(fi)bethecorrespondingpiecewisebilinearpol…     

变系数情形下Criss-Cross三角形线性元的渐近展式与超收敛

本文讨论了二阶椭圆方程变系数情形下Criss-Cross三角形线性元的超收敛性质,得到了有限元的渐进展式、外推及高精度组合公式等结果.     

SUPERCONVERGENCE ANALYSIS OF FINITE ELEMENT METHODS FOR OPTIMAL CONTROL PROBLEMS OF THE STATIONARY BENARD TYPE

In this paper, we consider the finite element approximation of the distributed optimal control problems of the stationary Benard type under the pointwise control constraint. The states and the co-states are approximated by polynomial functions of lowest-order mixed finite element space or piecewise linear functions and the control is approximated by piecewise constant functions. We give the superconvergence analysis for the control; it is proved that the approximation has a second-order rate of convergence. We further give the superconvergence analysis for the states and the co-states. Then we derive error estimates in L^∞-norm and optimal error estimates in L^2-norm.     

二阶常微分方程初值问题C^0有限元的超收敛

基于在一个单元上的改进的单元正交展开，得到二阶常微分方程C^0有限元Uh的误差表示式，以及节点和单元内部一些特征点上的超收敛结果。     

SUPERCONVERGENCE OF LEAST-SQUARES MIXED FINITE ELEMENT FOR SECOND-ORDER ELLIPTIC PROBLEMS

In this paper the least-squares mixed finite element is considered for solving secondorder elliptic problems in two dimensional domains. The primary solution u and the flux er are approximated using finite element spaces consisting of piecewise polynomials of degree k and r respectively. Based on interpolation operators and an auxiliary projection,superconvergent H^1-error estimates of both the primary solution approximation uh and the flux approximation σh are obtained under the standard quasi-uniform assumption on finite element partition. The superconvergence indicates an accuracy of O(h^r 2) for the least-squares mixed finite element approximation if Raviart-Thomas or Brezzi-DouglasFortin-Marini elements of order r are employed with optimal error estimate of O(h^r l).     

A SUPERONVERGENCE ANALYISIS FOR FINITE ELEMENT SOLUTION BY THE INTERPOLANT POSTPROCESSING ON IRREGULAR MESHES FOR SMOOTH PROBLEM

1. IntroductionThe results in this paper are based on the idea of interpolation postprocessing in [11 andthe techniques of LZ projection processing in [2]For simlicity) we consider the model problem: Finds e Hi(fl),such thatSuppose that Jh and JH are irregular triangulations (or quadrilateral partitions). Theirsizes satisfy h << H, (H - 0). Construct piecewise k-order and r-order finite element spaceSh and SH respectively. Let ah E Sh be the Galerkin approximation of u E HJ(fl), andbe …     

有限元Ritz-Volterra投影的超收敛性质及其应用

本文研究有限元Ritz－Volterra投影的超收敛性质．利用一种新型的Green函数，证明了该投影具有与有限元Ritz投影相平行的函数和导数逼近的超收敛性质．这些结果被应用于抛物型积分微分方程和Sobolev方程的半离散有限元近似．     

GLOBAL SUPERCONVERGENCE OF THE MIXED FINITEELEMENT METHODS FOR 2-D MAXWELL EQUATIONS

Superconvergence of the mixed finite element methods for 2-d Maxwell equations is studied in this paper. Two order of superconvergent factor can be obtained for the k-th Nedelec elements on the rectangular meshes.     

SUPERAPPROXIMATION PROPERTIES OF THE INTERPOLATION OPERATOR OF PROJECTION TYPE AND APPLICATIONS

AbstractSome superapproximation and ultra-approximation properties in function, gradient and two-order derivative approximations are shown for the interpolation operator of projection type on two-dimensional domain. Then, we consider the Ritz projection and Ritz-Volterra projection on finite element spaces, and by means of the superapproximation elementary estimates and Green function methods, derive the superconvergence and ultraconvergence error estimates for both projections, which are also the finite element approximation solutions of the elliptic problems and the Sobolev equations, respectively.     

有限元的一个局部超收敛结果

１．引 言 设 是一个有界开域，具充分光滑的边界 且设 是 上的一族拟一致的三角剖分，用 表示定义在Ｔｈ上的分片线性有限元空间，并置考虑模型问题 用 分别表示的有限元解及内插，那么有插值估计：（见［１］）一般地，如ｕ为问题（１．１）的解，我们有有限元逼近误差估计（见［３］） 命题１．设 并设 分别表示按定义的Ｇｒｅｅｎ函数及其有限元逼近，那么有其中 Ｃ与 ｚ，ｈ无关．（参见［３］） 注意．如 且 ，那么至少存在一个点 ，使即ｘ０是ｆ的奇点，例如其中 为常数， ，显然如果。，如果故我们假定 本文将证明，误差与ｆ的奇性…     

AN ANISOTROPIC NONCONFORMING FINITE ELEMENT WITH SOME SUPERCONVERGENCE RESULTS

The main aim of this paper is to study the error estimates of a nonconforming finite element with some superconvergence results under anisotropic meshes. The anisotropic interpolation error and consistency error estimates are obtained by using some novel approaches and techniques, respectively. Furthermore, the superclose and a superconvergence estimate on the central points of elements are also obtained without the regularity assumption and quasi-uniform assumption requirement on the meshes. Finally, a numerical test is carried out, which coincides with our theoretical analysis.     

FINITE VOLUME SUPERCONVERGENCE APPROXIMATION FOR ONE-DIMESIONAL SINGULARLY PERTURBED PROBLEMS

We analyze finite volume schemes of arbitrary order r for the one-dimensional singu- larly perturbed convection-diffusion problem on the Shishkin mesh. We show that the error under the energy norm decays as （N-11n（N 4- 1））r, where 2N is the number of subinter- vals of the primal partition. Furthermore, at the nodal points, the error in function value approximation super-converges with order （N-11n（N ＋ 1））2r, while at the Gauss points, the derivative error super-converges with order （N-11n（N ＋ 1））r＋1. All the above conver- gence and superconvergence properties are independent of the perturbation parameter e. Numerical results are presented to support our theoretical findings.     

ON THE ANISOTROPIC ACCURACY ANALYSIS OF ACM''''S NONCONFORMING FINITE ELEMENT

The main aim of this paper is to study the superconvergence accuracy analysis of thefamous ACM's nonconforming finite element for biharmonic equation under anisotropicmeshes. By using some novel approaches and techniques, the optimal anisotropic inter-polation error and consistency error estimates are obtained. The global error is of orderO(h~2). Lastly, some numerical tests are presented to verify the theoretical analysis.     

SUPERCONVERGENCE OF TETRAHEDRAL QUADRATIC FINITE ELEMENTS

For a model elhptic boundary value problem we will prove that on strongly regular families of uniform tetrahedral partitions of a pohyhedral domain, the gradient of the quadratic finite element approximation is superclose to the gradient of the quadratic La-grange interpolant of the exact solution. This supercloseness will be used to construct a post-processing that increases the order of approximation to the gradient in the global L^2-norm。