Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem

In this paper, we propose a method to improve the convergence rate of the lowest order Raviart-Thomas mixed finite element approximations for the second order elliptic eigenvalue problem. Here, we prove a supercloseness result for the eigenfunction approximations and use a type of finite element postprocessing operator to construct an auxiliary source problem. Then solving the auxiliary additional source problem on an augmented mixed finite element space constructed by refining the mesh or by using the same mesh but increasing the order of corresponding mixed finite element space, we can increase the convergence order of the eigenpair approximation. This postprocessing method costs less computation than solving the eigenvalue problem on the finer mesh directly. Some numerical results are used to confirm the theoretical analysis.     

Asymptotic expansion and extrapolation for the Eigenvalue approximation of the biharmonic eigenvalue problem by Ciarlet–Raviart scheme

We propose and analyze the Ciarlet–Raviart mixed scheme for solving the biharmonic eigenvalue problem with bilinear finite elements. We derive a higher order convergence rate for eigenvalue and eigenfunction approximations. Furthermore, we give an asymptotic expansion of the eigenvalue error, from which an efficient extrapolation and an a posteriori error estimate for the eigenvalue are given. Finally, numerical experiments illustrating the theoretical results are reported. This author was supported by China Postdoctoral Sciences Foundation.     

Approximation of an Eigenvalue Problem Associated with the Stokes Problem by the Stream Function-Vorticity-Pressure Method

By means of eigenvalue error expansion and integral expansion techniques, we propose and analyze the stream function-vorticity-pressure method for the eigenvalue problem associated with the Stokes equations on the unit square. We obtain an optimal order of convergence for eigenvalues and eigenfuctions. Furthermore, for the bilinear finite element space, we derive asymptotic expansions of the eigenvalue error, an efficient extrapolation and an a posteriori error estimate for the eigenvalue. Finally, numerical experiments are reported. The first author was supported by China Postdoctoral Sciences Foundation.     

用Fourier Galerkin方法求解间断特征值问题

本文我们考虑一个带间断系数的特征值问题,使用Fourier G a lerk in方法求解.数值试验表明特征值收敛速度达到三阶,而特征函数收敛速度达到2.5阶.表明此方法对间断系数问题非常有效.     

Approximation of sign-indefinite spectral problems

A variational sign-indefinite eigenvalue problem in an infinite-dimensional Hilbert space is approximated by a problem in a finite-dimensional subspace. We analyze the convergence and accuracy of approximate eigenvalues and eigenelements. The general results are illustrated by a sample scheme of the finite-element method with numerical integration for a one-dimensional sign-indefinite second-order differential eigenvalue problem.     

A Zienkiewicz-type finite element applied to fourth-order problems

This paper deals with convergence analysis and applications of a Zienkiewicz-type (Z-type) triangular element, applied to fourth-order partial differential equations. For the biharmonic problem we prove the order of convergence by comparison to a suitable modified Hermite triangular finite element. This method is more natural and it could be applied to the corresponding fourth-order eigenvalue problem. We also propose a simple postprocessing method which improves the order of convergence of finite element eigenpairs. Thus, an a posteriori analysis is presented by means of different triangular elements. Some computational aspects are discussed and numerical examples are given.     

Approximation of variational eigenvalue problems

A variational eigenvalue problem in an infinite-dimensional Hilbert space is approximated by a problem in a finite-dimensional subspace. We analyze the convergence and accuracy of the approximate solutions. The general results are illustrated by a scheme of the finite element method with numerical integration for a one-dimensional second-order differential eigenvalue problem. For this approximation, we obtain optimal estimates for the accuracy of the approximate solutions.     

FD-method for an eigenvalue problem with nonlinear potential

Using the functional discrete approach and Adomian polynomials, we propose a numerical algorithm for an eigenvalue problem with a potential that consists of a nonlinear autonomous part and a linear part depending on an independent variable. We prove that the rate of convergence of the algorithm is exponential and improves as the order number of an eigenvalue increases. We investigate the mutual influence of the piecewise-constant approximation of the linear part of the potential and the nonlinearity on the rate of convergence of the method. Theoretical results are confirmed by numerical data. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 14–28, January, 2007.     

ON THE DISCRETE-ORDINATES METHOD FOR THE MONOENERGETIC NEUTRON TRANSPORT EQUATION IN A SLAB WITH GENERALIZED BOUNDARY CONDITIONS

This paper is devoted to discussing the discrete-ordinates method for the monoenergetic neutron transport equation in a slab with generalized boundary conditions. For homogeneous medium with isotropic scattering and fission, the convergence theorems for discrete-ordinates approximations are given respectively for critical eigenvalue problem and dominant eigenvalue problems: for inhomogeneous medium with anisotropic scattering and fission, a similar discussion and an estimation for the convergence rate are given for critical eigenvalue problems. Finally, some numerical results are given by use of this method.     

Extremal Eigenvalues of the Sturm-Liouville Problems with Discontinuous Coefficients

In this paper, an extremal eigenvalue problem to the Sturm-Liouville equations with discontinuous coefficients and volume constraint is investigated. Liouville transformation is applied to change the problem into an equivalent minimization problem. Finite element method is proposed and the convergence for the finite element solution is established. A monotonic decreasing algorithm is presented to solve the extremal eigenvalue problem. A global convergence for the algorithm in the continuous case is proved. A few numerical results are given to depict the efficiency of the method.     

Das Differenzenverfahren für singuläre Rand-Eigenwertaufgaben gewöhnlicher Differentialgleichungen

Summary The boundary value problem for a class of singular second order differential operators is defined. Using the standard three point discretisation for the differential equation and taking care of the limits involved in the boundary conditions in a natural way, finite difference approximations to the boundary value problems are defined and their convergence properties are investigated. The rate of convergence is given in terms of the data. It turns out that for problems of the first kind extrapolation is possible up to an arbitrary order after a suitable change of the independent variable, whereas for problems of the second kind neither theoretical nor numerical results indicate the possibility of extrapolation. Corresponding results hold for the eigenvalue problems. Some numerical examples show that the convergence rates given in the paper are best possible and demonstrate the effect of extrapolation.     

A boundary element method for the Dirichlet eigenvalue problem of the Laplace operator

The solution of eigenvalue problems for partial differential operators by using boundary integral equation methods usually involves some Newton potentials which may be resolved by using a multiple reciprocity approach. Here we propose an alternative approach which is in some sense equivalent to the above. Instead of a linear eigenvalue problem for the partial differential operator we consider a nonlinear eigenvalue problem for an associated boundary integral operator. This nonlinear eigenvalue problem can be solved by using some appropriate iterative scheme, here we will consider a Newton scheme. We will discuss the convergence and the boundary element discretization of this algorithm, and give some numerical results.     

Newton's Method for a Generalized Inverse Eigenvalue Problem

A kind of generalized inverse eigenvalue problem is proposed which includes the additive, multiplicative and classical inverse eigenvalue problems as special cases. Newton's method is applied, and a local convergence analysis is given for both the distinct and the multiple eigenvalue cases. When the multiple eigenvalues are present we show how to state the problem so that it is not over-determined, and discuss a Newton-method for the modified problem. We also prove that the modified method retains quadratic convergence, and present some numerical experiments to illustrate our results. © 1997 by John Wiley & Sons, Ltd.     

Greedy Algorithms for High-Dimensional Eigenvalue Problems

In this article, we present two new greedy algorithms for the computation of the lowest eigenvalue (and an associated eigenvector) of a high-dimensional eigenvalue problem and prove some convergence results for these algorithms and their orthogonalized versions. The performance of our algorithms is illustrated on numerical test cases (including the computation of the buckling modes of a microstructured plate) and compared with that of another greedy algorithm for eigenvalue problems introduced by Ammar and Chinesta.     

广义块Toeplitz特征值问题的基于sine变换的预处理子

在求块Toeplitz矩阵束（Amn，Bmn）特征值的Lanczos过程中，通过对移位块Toepltz矩阵Amn-ρBmn进行基于sine变换的块预处理，从而改进了位移块Toeplitz矩阵的谱分布，加速了Lanczos过程的收敛速度．该块预处理方法能通过快速算法有效快速执行．本文证明了预处理后Lanczos过程收敛迅速，并通过实验证明该算法求解大规模矩阵问题尤其有效．     

Existence and approximation results for shape optimization problems in rotordynamics

We consider a shape optimization problem in rotordynamics where the mass of a rotor is minimized subject to constraints on the natural frequencies. Our analysis is based on a class of rotors described by a Rayleigh beam model including effects of rotary inertia and gyroscopic moments. The solution of the equation of motion leads to a generalized eigenvalue problem. The governing operators are non-symmetric due to the gyroscopic terms. We prove the existence of solutions for the optimization problem by using the theory of compact operators. For the numerical treatment of the problem a finite element discretization based on a variational formulation is considered. Applying results on spectral approximation of linear operators we prove that the solution of the discretized optimization problem converges towards the solution of the continuous problem if the discretization parameter tends to zero. Finally, a priori estimates for the convergence order of the eigenvalues are presented and illustrated by a numerical example.     

解Stokes特征值问题的一种两水平稳定化有限元方法

基于局部Gauss积分,研究了解Stokes特征值问题的一种两水平稳定化有限元方法．该方法涉及在网格步长为H的粗网格上解一个Stokes特征值问题,在网格步长为h=O(H2)的细网格上解一个Stokes问题．这样使其能够仍旧保持最优的逼近精度,求得的解和一般的稳定化有限元解具有相同的收敛阶,即直接在网格步长为h的细网格上解一个Stokes特征值问题．因此,该方法能够节省大量的计算时间．数值试验验证了理论结果．     

Error bounds and estimates for eigenvalues of integral equations

Summary Approximate solutions of the linear integral equation eigenvalue problem can be obtained by the replacement of the integral by a numerical quadrature formula and then collocation to obtain a linear algebraic eigenvalue problem. This method is often called the Nyström method and its convergence was discussed in [7]. In this paper computable error bounds and dominant error terms are derived for the approximation of simple eigenvalues of nonsymmetric kernels.     

基于形状优化框架下的Steklov特征值问题研究

<正>1引言特征值问题在应用数学分支和工程中,尤其是在最优设计问题中,有很多的应用,所以特征值问题的最优化已经有了较为深入的研究,见在我们的研究当中,最优设计问题常常以一种指定载荷的设计下、能量的极小化问题的形式出现.在大多数关于最优设计的文章里面,我们更重视在一个固定载荷下条件下结构的最     

Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions

By taking as a “prototype problem” a one-delay linear autonomous system of delay differential equations we present the problem of computing the characteristic roots of a retarded functional differential equation as an eigenvalue problem for a derivative operator with non-local boundary conditions given by the particular system considered. This theory can be enlarged to more general classes of functional equations such as neutral delay equations, age-structured population models and mixed-type functional differential equations.It is thus relevant to have a numerical technique to approximate the eigenvalues of derivative operators under non-local boundary conditions. In this paper we propose to discretize such operators by pseudospectral techniques and turn the original eigenvalue problem into a matrix eigenvalue problem. This approach is shown to be particularly efficient due to the well-known “spectral accuracy” convergence of pseudospectral methods. Numerical examples are given.