基于非协调有限元方法的特征值的下界逼近

本文将文献提出的非协调元方法用于二阶椭圆特征值问题,证明了最优的误差估计.并且证明了当网格充分细时,近似特征值总是比真解小.     

Wilson元特征值下逼近准确特征值

该文讨论矩形域上Laplace算子特征值问题有限元近似.证明了Wilson非协调有限元特征值下逼近准确特征值,从而解决了有限元法中长期存在的一个猜想.     

双曲型方程的质量集中有限元法

用有限元法求解波动方程有许多工作,在此不多加阐述,本文针对线性双曲型方程采用质量集中有限元法,该方法产生于用特定的数值积分公式计算普通有限元法中的内积积分,这种方法具有较好的数值稳定性。     

UNIFORM SUPERAPPROXIMATION OF THE DERIVATIVE OF TETRAHEDRAL QUADRATIC FINITE ELEMENT APPROXIMATION

In this paper,we will prove the derivative of tetrahedral quadratic finite element ap-proximation is superapproximate to the derivative of the quadratic Lagrange interpolant of the exact solution in the L^∞-norm, which can be used to enhance the accuracy of the derivative of tetrahedral quadratic finite element approximation to the derivative of the exact solution。     

FINITE ELEMENT APPROXIMATION OF A NONLINEAR STEADY-STATE HEAT CONDUCTION PROBLEM

1. Introduction Anciellt Chinese mathematicians have done many fund~al discoveries, even thoughmad of them are now usually called by western names, e.g., the Pascal triangle, the HornerIcheme, the Gaussian elimination, see [231. In the modern era3 Chinese mathematiciare have.jalso got many priorities. For instance, the fast Proof Of convergence of the finite element methodfor a linear elliptic boundary vajue problem was done in the pioneering work [61 by K. Feng in1965 (for the English tra…     

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…     

ON THE ERROR ESTIMATE OF LINEAR FINITE ELEMENT APPROXIMATION TO THE ELASTIC CONTACT PROBLEM WITH CURVED CONTACT BOUNDARY

1. PreliminaryThe error estimate of linear finite element approximation to the elastic contact problem with curved contact boundary was considered in [2], in which the authors obtainedthe error bound of O(hl) with a much complex proof, requirement of three times continuously differentiable for contact boundary and extra regularities of triangulation (c.f. [2, Theorem 3.3, p.149]). In this paper, we obtained the error bound of O(hi) withonly requirement of two times continuously differentiable…     

特征值问题混合有限元法的一个误差估计

设（λh,σh,uh）是一个混合有限元特征对.Babuska和Osborn建立了（λh,uh）的误差估计.本文导出了σh的抽象误差估计式.并把该估计式应用于二阶椭圆特征值问题Raviart-Thomas混合有限元格式和重调和算子特征值问题Ciarlet-Raviart混合有限元格式,得到了一些新的误差估计.     

杆有限元模型的特征值反问题

1引言非均匀杆的轴向自由振动方程为其中L为杆的长度,A(x)为杆的横截面积,E(x)为弹性模量,ρ(x)是杆的密度．杆的主振动的一般形式为u(x,t)=u(x)sin(wt φ), (2)其中u(x)满足d/dx[E(x)A(x)du/dx] λρ(x)A(x)u=0,λ=ω2．(3)     

ON THE FINITE ELEMENT APPROXIMATION OF SYSTEMS OF REACTION-DIFFUSION EQUATIONS BY p/hp METHODS

We consider the approximation of systems of reaction-diffusion equations, with the finite element method. The highest derivative in each equation is multiplied by a parameter ε∈ （0, 1], and as ε → 0 the solution of the system will contain boundary layers. We extend the analysis of the corresponding scalar problem from [Melenk, IMA J. Numer. Anal. 17（1997）, pp. 577-601], to construct a finite element scheme which includes elements of size O（εp） near the boundary, where p is the degree of the approximating polynomials. We show that, under the assumption of analytic input data, the method yields exponential rates of convergence, independently of ε, when the error is measured in the energy norm associated with the problem. Numerical computations supporting the theory are also presented, which also show that the method yields robust exponential convergence rates when the error in the maximum norm is used.     

SUPER-GEOMETRIC CONVERGENCE OF A SPECTRAL ELEMENT METHOD FOR EIGENVALUE PROBLEMS WITH JUMP COEFFICIENTS

We propose and analyze a C^0 spectral element method for a model eigenvalue problem with discontinuous coefficients in the one dimensional setting. A super-geometric rate of convergence is proved for the piecewise constant coefficients case and verified by numerical tests. Furthermore, the asymptotical equivalence between a Gauss-Lobatto collocation method and a spectral Galerkin method is established for a simplified model.     

一类变分不等方程的有限元逼近

本文对一类带有两个变函数的非线性变分不等方程进行讨论，所讨论的区域Ω＝Ｒ＾ｎ是有界子开子集，边界具有分片Ｃ＾２光滑，但不必为凸区域，文中给出了解的有限元逼近及逼近的误差估计。     

GLOBAL FINITE ELEMENT NONLINEAR GALERKIN METHOD FOR THE PENALIZED NAVIER-STOKES EQUATIONS GLOBAL FINITE ELEMENT NONLINEAR GALERKIN METHOD FOR THE PENALIZED NAVIER-STOKES EQUATIONS

1. IntroductionIn the numerical simulation of the Navier-Stokes equations one encounters three seriousdifficulties in the case of large Reynolds numbers f the treatment of the incomPressibility con-dition divu = 0, the treatment of the noIilinear terms and the large time integration. For thetreatment of the incoInPressibility condition, one use the penalty method in the case of finiteelemellts [1--2l and for the treatmen of the noulinar terms and the large tfor integration, oneuse the nonlin…     

TIME-DISCRETIZATION PROCEDURE FOR FINITE ELEMENT APPROXIMATION OF COMPRESSIBLE MISCIBLE DISPLACEMENT IN POROUS MEDIA

We'll consider the model of two-phase compressible miscible displacement in porous media which includes molecular diffusion and dispersion in one dimensional space. Time-discretization procedure is established and analysed. The optimal error estimate in L2 norm is proved by introducing a new interpolation operator R.     

PRECONDITIONING HIGHER ORDER FINITE ELEMENT SYSTEMS BY ALGEBRAIC MULTIGRID METHOD OF LINEAR ELEMENTS

We present and analyze a robust preconditioned conjugate gradient method for the higher order Lagrangian finite element systems of a class of elliptic problems. An auxiliary linear element stiffness matrix is chosen to be the preconditioner for higher order finite elements. Then an algebraic multigrid method of linear finite element is applied for solving the preconditioner. The optimal condition number which is independent of the mesh size is obtained. Numerical experiments confirm the efficiency of the algorithm.     

MULTIGRID ALGORITHM FOR THE COUPLING SYSTEM OF NATURAL BOUNDARY ELEMENT METHOD AND FINITE ELEMENT METHOD FOR UNBOUNDED DOMAIN PROBLEMS

In this paper, some V-cycle multigrid algorithms are presented for the coupling system arising from the discretization of the Dirichlet exterior problem by coupling the natural boundary element method and finite element method. The convergence of these multigrid algorithms is obtained even with only one smoothing on all levels. The rate of convergence is found uniformly bounded independent of the number of levels and the mesh sizes of all levels, which indicates that these multigrid algorithms are optimal. Some numerical results are also reported.     

ADAPTIVE FINITE ELEMENT APPROXIMATION FOR A CLASS OF PARAMETER ESTIMATION PROBLEMS

In this paper, we study adaptive finite element discretisation schemes for a class of parameter estimation problem. We propose to efficient algorithms for the estimation problem use adaptive multi-meshes in developing We derive equivalent a posteriori error estimators for both the state and the control approximation, which particularly suit an adaptive multi-mesh finite element scheme. The error estimators are then implemented and tested with promising numerical results.     

CONVERGENCE AND SUPERCONVERGENCE OF HERMITE BICUBIC ELEMENT FOR EIGENVALUE PROBLEM OF THE BIHARMONIC EQUATION

Eigenvalue problem for biharmonic equation is an interesting and important problem, seeCiarlet and Lions[3]. In 1979, Rannacher[8] used the Adini nonconforming finite element tosolve this problem and obtained:Recedely, Yang[6] has proved that the order of covergence of Ah is just 2. The aim of this paperis to improve the order of convergence by using Hermite bicubic element. To our knowledge,there is not any result for approximation to the eigenvalue problem by using this element inliteratu…     

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…     

CONVERGENCE OF AN EXPLICIT UPWIND FINITE ELEMENT METHOD TO MULTI-DIMENSIONAL CONSERVATION LAWS

1. IntroductionThe convergence problem for the numerical schemes to one dimensional cons~inn lawshas been edensively studied. By tensor product one dimensional schemes can be appliedto multi-dimensional equations. However the convergence of many of those schemes is stillchanknown even if it is true for one dimensional cases. Besides, for those Phys.ical domains withcomplicated geometry unstructured grids are more fiekible. In recent yeaes the convergellceproblem for unstructured grids has cal…