三维区域上Q_1~(rot)元的渐近展开及外推

本文研究了正方体区域上Q₁^rot非协调元渐近展开式.利用林群、吕涛等提出的有限元误差渐近展开法,获得了正方体区域上Q₁^rot非协调元特征值的误差渐近展开式.理论分析和数值实验结果表明三维Q₁^rot非协调元特征值外推公式是有效的,可以把特征值的精度从二阶提高到四阶.     

三维Poisson方程特征值的四种有限元解及比较

应用三维EQ1rot元、三维Crouzeix-Raviart元、八节点等参数元、四面体线性元计算三维Poisson方程的近似特征值.计算结果表明:三维EQ1rot元和三维Crouzeix-Raviart元特征值下逼近准确特征值,八节点等参数元、四面体线性元特征值上逼近准确特征值,三维EQr1ot元和三维Crouzeix-Raviart元外推特征值下逼近准确特征值.计算结果还表明三维Crouzeix-Raviart元是一种计算效率较高的非协调元.     

特征值问题协调/非协调有限元后验误差分析

本文研究对称椭圆特征值问题的有限元后验误差估计,包括协调元和非协调元,具有下列特色:(1)对协调/非协调元建立了有限元特征函数uh的误差与相应的边值问题有限元解的误差在局部能量模意义下的恒等关系式,该边值问题的右端为有限元特征值λh与uh的乘积,有限元解恰好为uh.从而边值问题有限元解在能量模意义下的局部后验误差指示子,包括残差型和重构型后验误差指示子,成为有限元特征函数在能量模意义下的局部后验误差指示子.(2)讨论了协调有限元特征函数的基于插值后处理的梯度重构型后验误差估计,对有限元特征函数的导数得到了最大模意义下的渐近准确局部后验误差指示子.     

Equality of multiplicities of a Sturm-Liouville eigenvalue

We give a new and unified proof of the fact that for any eigenvalue of a self-adjoint Sturm-Liouville problem with limit-circle end points, its analytic and geometric multiplicities are equal. This proof is geometric in nature and can be generalized to the case of high-order differential equations.     

A note on positive eigenfunctions and hidden convexity

We give a simple convexity-based proof of the following fact: the only eigenfunction of the p-Laplacian that does not change sign is the first one. The method of proof covers also more general nonlinear eigenvalue problems.     

Stability estimates for h-p spectral element methods for general elliptic problems on curvilinear domains

In this paper we show that the h-p spectral element method developed in [3,8,9] applies to elliptic problems in curvilinear polygons with mixed Neumann and Dirichlet boundary conditions provided that the Babuska-Brezzi inf-sup conditions are satisfied. We establish basic stability estimates for a non-conforming h-p spectral element method which allows for simultaneous mesh refinement and variable polynomial degree. The spectral element functions are non-conforming if the boundary conditions are Dirichlet. For problems with mixed boundary conditions they are continuous only at the vertices of the elements. We obtain a stability estimate when the spectral element functions vanish at the vertices of the elements, which is needed for parallelizing the numerical scheme. Finally, we indicate how the mesh refinement strategy and choice of polynomial degree depends on the regularity of the coefficients of the differential operator, smoothness of the sides of the polygon and the regularity of the data to obtain the maximum accuracy achievable. An erratum to this article is available at .     

Discrete compactness and the approximation of Maxwell's equations in

We analyze the use of edge finite element methods to approximate Maxwell's equations in a bounded cavity. Using the theory of collectively compact operators, we prove -convergence for the source and eigenvalue problems. This is the first proof of convergence of the eigenvalue problem for general edge elements, and it extends and unifies the theory for both problems. The convergence results are based on the discrete compactness property of edge element due to Kikuchi. We extend the original work of Kikuchi by proving that edge elements of all orders possess this property.

     

广义协调等参元

本文根据广义协调条件,对平面应力四边形单元提出一个广义协调等参元GC-Q6.单元GC-Q6是对Wilson非协调等参元Q6的一个改进方案:单元Q6只对平行四边形网格情况能通过分片检验,对一般四边形情况却不能通过,而本文的单元GC-Q6则对一般四边形情况也能通过.当单元为平行四边形时,GC-Q6单元即退化为QS单元.算例表明,广义协调等参元的应力精度高于文献中已有的单元,对不规则网格均能保持良好的性态.     

关于单李代数的Coxeter-Killing变换

<正> 决定复单李群(代数)的Betti数是个经典的问题.大家知道,它们就是该李群(代数)的Poincare多项式     

A posteriori error estimators for the Stokes equations II non-conforming discretizations

Summary We present an a posteriori error estimator for the non-conforming Crouzeix-Raviart discretization of the Stokes equations which is based on the local evaluation of residuals with respect to the strong form of the differential equation. The error estimator yields global upper and local lower bounds for the error of the finite element solution. It can easily be generalized to the stationary, incompressible Navier-Stokes equations and to other non-conforming finite element methods. Numerical examples show the efficiency of the proposed error estimator.     

Approximation of functionally graded plates with non-conforming finite elements

In this paper rectangular plates made of functionally graded materials (FGMs) are studied. A two-constituent material distribution through the thickness is considered, varying with a simple power rule of mixture. The equations governing the FGM plates are determined using a variational formulation arising from the Reissner–Mindlin theory. To approximate the problem a simple locking-free Discontinuous Galerkin finite element of non-conforming type is used, choosing a piecewise linear non-conforming approximation for both rotations and transversal displacement. Several numerical simulations are carried out in order to show the capability of the proposed element to capture the properties of plates of various gradings, subjected to thermo-mechanical loads.     

A high-order non-conforming discontinuous Galerkin method for time-domain electromagnetics

In this paper, we discuss the formulation, stability and validation of a high-order non-dissipative discontinuous Galerkin (DG) method for solving Maxwell’s equations on non-conforming simplex meshes. The proposed method combines a centered approximation for the numerical fluxes at inter element boundaries, with either a second-order or a fourth-order leap-frog time integration scheme. Moreover, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary-level hanging nodes. The method is proved to be stable and conserves a discrete counterpart of the electromagnetic energy for metallic cavities. Numerical experiments with high-order elements show the potential of the method.     

关于非协调位移元与杂交应力元的对应性

本文阐明了E.L.Wilson[3]等的非协调移位元与卞学鐄的杂交应力元之间所存在的对应性.     

Richard courant and the finite element method: A further look

The finite element method has in recent years become one of the most popular and effective numerical procedures for boundary value and eigenvalue problems. In the early 1960s it was found that the method was an independent rediscovery of a simpler idea proposed in 1943 by Richard Courant. The importance of Courant's 1943 paper has been emphasized by G. Strang in 1973 and by others. This note seeks to describe briefly Courant's finite element work which led to his publication of 1943. It is shown that Courant used the finite element ideas as early as 1922 in a proof employing Dirichlet's principle.     

On mappings whose sum is monotone

Proposition 1 of this article points at a gap in the proof of Kuhlmann’s characterization of algebraically maximal valued fields [2]. The author [1] showed how to fill this gap. His arguments involved tools of mathematical logic and infinite combinatorics. Proposition 2 of this article provides us with a simple proof of the key fact of Kuhlmann’s characterization.     

非协调区域分解的Lagrangian乘子法

§1.引言 近年来随着并行计算机的迅速发展,求解椭圆型方程的区域分解法愈来愈引起人们的兴趣和重视.但是,目前能够见到的有限元区域分解法几乎都要求有限元空间在跨过子区域的边界时是协调的,必然限制有限元区域分解算法的优越性. [3]提出了一种非协凋区域分解法——非协调区域分解的杂交法.采用简化杂交法处理各子区域交界处的非协调性,这种方法在子区域的内部和边界采用两套不同的变量,允许内部变量在跨过各子区域的边界时不连续.但是这种方法有它的局限性,即要求边界变量在各子区域的顶点处必须保持连续性,这对推广到三维空间的情形带来很大的困难.本文提出一种非协调区域分解的Lagrangian乘子法,引进Lagrangian乘子来处理各子区域交界处的非协调性.这种方法也在子区域内部和边界采用两套不同的变量,它不仅允许内部变量在越过各子区域边界时的非协调性,并且还允许边界变量在各子区域的顶点处可以不连续,这就弥补了[3]的不足.同时,这种算法具有[3]的优点,即在不     

基于Crouzeix-Raviart元的界面浸入有限元方法及其收敛性分析

本文对具间断系数的二阶椭圆界面问题提出一种浸入有限元方法(theimmersed finite element method), 即在界面单元上采用依赖于界面的线性多项式空间离散, 而在非界面单元上采用Crouzeix-Raviart非协调元离散. 论证表明, 该方法具有对界面问题解的最优L²-模和H¹-模收敛精度.     

A framework for obtaining guaranteed error bounds for finite element approximations

We give an overview of our recent progress in developing a framework for the derivation of fully computable guaranteed posteriori error bounds for finite element approximation including conforming, non-conforming, mixed and discontinuous finite element schemes. Whilst the details of the actual estimator are rather different for each particular scheme, there is nonetheless a common underlying structure at work in all cases. We aim to illustrate this structure by treating conforming, non-conforming and discontinuous finite element schemes in a single framework. In taking a rather general viewpoint, some of the finer details of the analysis that rely on the specific properties of each particular scheme are obscured but, in return, we hope to allow the reader to ‘see the wood despite the trees’.     

A proof of the Faber-Krahn inequality for the first eigenvalue of thep-Laplacian

Summary In this work we present an elementary proof of the Faber-Krahn inequality for the first eigenvalue of the p-Laplacian on bounded domains in ℝⁿ. Let λ₁ be the first eigenvalue and λ ₁ ^* be the first eigenvalue for the ball of the same volume. Then we show that λ₁=λ ₁ ^* iff the domain is a ball. Our proof makes considerable use of the corresponding Talenti's inequality and some well known properties of the first eigenfunction. Entrata in Redazione il 20 aprile 1998.     

Steklov-Neumann eigenproblems and nonlinear elliptic equations with nonlinear boundary conditions

We study the solvability of nonlinear second order elliptic partial differential equations with nonlinear boundary conditions. We introduce the notion of “eigenvalue-lines” in the plane; these eigenvalue-lines join each Steklov eigenvalue to the first eigenvalue of the Neumann problem with homogeneous boundary condition. We prove existence results when the nonlinearities involved asymptotically stay, in some sense, below the first eigenvalue-lines or in a quadrilateral region (depicted in Fig. 1) enclosed by two consecutive eigenvalue-lines. As a special case we derive the so-called nonresonance results below the first Steklov eigenvalue as well as between two consecutive Steklov eigenvalues. The case in which the eigenvalue-lines join each Neumann eigenvalue to the first Steklov eigenvalue is also considered. Our method of proof is variational and relies mainly on minimax methods in critical point theory.