A POSTERIORI ESTIMATOR OF NONCONFORMING FINITE ELEMENT METHOD FOR FOURTH ORDER ELLIPTIC PERTURBATION PROBLEMS

In this paper, we consider the nonconforming finite element approximations of fourth order elliptic perturbation problems in two dimensions. We present an a posteriori error estimator under certain conditions, and give an h-version adaptive algorithm based on the error estimation. The local behavior of the estimator is analyzed as well. This estimator works for several nonconforming methods, such as the modified Morley method and the modified Zienkiewicz method, and under some assumptions, it is an optimal one. Numerical examples are reported, with a linear stationary Cahn-HiUiard-type equation as a model problem.     

SUPERCONVERGENCE ANALYSIS OF THE STABLE CONFORMING RECTANGULAR MIXED FINITE ELEMENTS FOR THE LINEAR ELASTICITY PROBLEM

In this paper, we consider the linear elasticity problem based on the Hellinger-Reissner variational principle. An O（h2） order superclose property for the stress and displacement and a global superconvergence result of the displacement are established by employing a Clement interpolation, an integral identity and appropriate postprocessing techniques.     

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.     

SUPERCONVERGENCE OF GRADIENT RECOVERY SCHEMES ON GRADED MESHES FOR CORNER SINGULARITIES

For the linear finite element solution to the Poisson equation, we show that supercon- vergence exists for a type of graded meshes for corner singularities in polygonal domains. In particular, we prove that the L^2-projection from the piecewise constant field △↓UN to the continuous and piecewise linear finite element space gives a better approximation of △↓U in the Hi-norm. In contrast to the existing superconvergence results, we do not assume high regularity of the exact solution.     

UNIFORM SUPERCONVERGENCE OF GALERKIN METHODS FOR SINGULARLY PERTURBED PROBLEMS

In this paper, we are concerned with uniform superconvergence of Galerkin methods for singularly perturbed reaction-diffusion problems by using two Shishkin-type meshes. Based on an estimate of the error between spline interpolation of the exact solution and its numerical approximation, an interpolation post-processing technique is applied to the original numerical solution. This results in approximation exhibit superconvergence which is uniform in the weighted energy norm. Numerical examples are presented to demonstrate the effectiveness of the interpolation post-processing technique and to verify the theoretical results obtained in this paper.     

Two order superconvergence of finite element methods for Sobolev equations

We consider finite element methods applied to a class of Sobolev equations inR ^d(d ≥ 1). Global strong superconvergence, which only requires that partitions are quais-uniform, is investigated for the error between the approximate solution and the Ritz-Sobolev projection of the exact solution. Two order superconvergence results are demonstrated inW ^1,p (Ω) andL _p(Ω) for 2 ≤p < ∞.     

Carleson-Type Maximal Operators with Variable Kernels

The author considers the L^p boundedness for two kinds of Carleson-type maximal operators with variable kernels Ω（x,y＇）/｜y｜^n,where Ω（x,y＇）∈L^∞（R^n）×W2^s（S^n-1）for some s〉0.     

Existence and Uniqueness of Renormalized Solution of Nonlinear Degenerated Elliptic Problems

In this paper, We study a general class of nonlinear degenerated elliptic problems associated with the differential inclusion β(u)-div(a(x, Du)+ F(u)) ■ f in Ω, where f ∈ L1 Ω. A vector field a(·,·) is a Carath′eodory function. Using truncation techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for general L1-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established.     

Lower bound for quantum integration error on anisotropic Sobolev classes

We study the approximation of the integration of multivariate functions in the quantum model of computation. Using a new reduction approach we obtain a lower bound of the n-th minimal query error on anisotropic Sobolev class R（Wpr（[0, 1]d）） （r R＋d）. Then combining this result with our previous one we determine the optimal bound of n-th minimal query error for anisotropic Hblder- Nikolskii class R（H∞r（[0,1]d）） and Sobolev class R（W∞r（[0,1]d））. The results show that for these two types of classes the quantum algorithms give significant speed up over classical deterministic and randomized algorithms.     

Operator Equations and Duality Mappings in Sobolev Spaces with Variable Exponents

After studying in a previous work the smoothness of the space UΓ0={u∈W1,p(·)(Ω);u=0 on Γ0 Γ=Ω},where dΓ-measΓ0>0,with p(·)∈C(Ω)and p(x)>1 for all x∈Ω,the authors study in this paper the strict and uniform convexity as well as some special properties of duality mappings defined on the same space.The results obtained in this direction are used for proving existence results for operator equations having the form Ju=Nfu,where J is a duality mapping on UΓ0 corresponding to the gauge function,and Nf is the Nemytskij operator generated by a Carath′eodory function f satisfying an appropriate growth condition ensuring that Nf may be viewed as acting from UΓ0 into its dual.     

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．     

Superconvergence analysis for Maxwell's equations in dispersive media

In this paper, we consider the time dependent Maxwell's equations in dispersive media on a bounded three-dimensional domain. Global superconvergence is obtained for semi-discrete mixed finite element methods for three most popular dispersive media models: the isotropic cold plasma, the one-pole Debye medium, and the two-pole Lorentz medium. Global superconvergence for a standard finite element method is also presented. To our best knowledge, this is the first superconvergence analysis obtained for Maxwell's equations when dispersive media are involved.

     

有限元Ritz-Volterra投影的插值后处理

１．引言 有限元后验误差估计和超收敛性质在有限元计算中具有重要的实际意义．近年来,这方面的研究工作已取得较丰富的研究结果［１－４］,其中林群等提出的有限元插值后处理技术是一种很有效的研究手段．但目前的已有结果主要是关于椭圆问题有限元近似．本文将研究与时间依赖问题有限元方法密切相关的有限元 Ｒｉｔｚ－ Ｖｏｌｔｅｒｒａ投影问,在一些超收敛估计的基础上,利用插值后处理技术,得到了该投影经插值后处理后在 Ｌ２, Ｈ１, Ｌ∞和 Ｗ∞１范数下的整体超收敛性,进而导出在相应范数下的渐进准确后验误差估计．这些结果…     

A direct global superconvergence analysis for Sobolev and viscoelasticity type equations

In this paper we study the finite element approximations to the Sobolev and viscoelasticity type equations and present a direct analysis for global superconvergence for these problems, without using Ritz projection or its modified forms.     

FINITE ELEMENT METHODS FOR SOBOLEV EQUATIONS

A new high-order time-stepping finite element method based upon the high-order numerical integration formula is formulated for Sobolev equations, whose computations consist of an iteration procedure coupled with a system of two elliptic equations. The optimal and superconvergence error estimates for this new method are derived both in space and in time. Also, a class of new error estimates of convergence and superconvergence for the time-continuous finite element method is demonstrated in which there are no time derivatives of the exact solution involved, such that these estimates can be bounded by the norms of the known data. Moreover, some useful a-posteriori error estimators are given on the basis of the superconvergence estimates.     

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.     

非线性常微分方程初值问题的间断有限元

该文用m次间断有限元求解非线性常微分方程初值问题u'=f(x,u),u(0)=u₀,用单元正交投影及正交性质证明了当m≥1时,m次间断有限元在节点x_j的左极限U(x_j-0)有超收敛估计(u-U(x_j-0)=O(h^2m+1),在每个单元内的m+1阶特征点x_ji上有高一阶的超收敛性(u-U)(x_ji)=O(h^m+2).     

Sobolev方程各向异性矩形非协调有限元分析

研究了Sobolev方程的各向异性矩形非协调有限元方法．在半离散和全离散格式下,得到了与传统协调有限元方法相同的最优误差估计和超逼近性质．进一步地利用插值后处理技术得到了整体超收敛结果．最后的数值结果表明了理论分析的正确性．     

常微方程组初值问题的连续有限元及守恒性

In this paper the continous finite element to solve initinal value problem for system of linear differential equations is used, and the absolute stability of the corresponding single step k-order hidden shceme is discussed. In the paper by simplified means, the superconvergence of finite element and one of it on the nodes are proved. Using the continuous finite element to solve linear Hamilton systems: Pt = Hq,qt = -Hp, the conservation of energy H(p,q) = 1/2 ap^2 bpq 1/2 cq^2 can be obtained. The computation shows that even if division is regular and the error of finite element Ph, qh is big, H(ph, qh) is almost equal H(p, q) in the range of computation accuracy.     

NONLINEAR RANK-ONE MODIFICATION OF THE SYMMETRIC EIGENVALUE PROBLEM

Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In this paper, we first study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method （SLAM）. The global convergence of the SLAM is proven under some mild assumptions. Numerical examples illustrate that the SLAM is the most robust method.