The extrapolation of numerical eigenvalues by finite elements for differential operators

This paper discusses the extrapolation of numerical eigenvalues by finite elements for differential operators and obtains the following new results: (a) By extending a theorem of eigenvalue error estimate, which was established by Osborn, a new expansion of eigenvalue error is obtained. Many achievements, which are about the asymptotic expansions of finite element methods of differential operator eigenvalue problems, are brought into the framework of functional analysis. (b) The Richardson extrapolation of nonconforming finite elements for multiple eigenvalues and splitting extrapolation of finite elements based on domain decomposition of non-selfadjoint differential operators for multiple eigenvalues are achieved. In addition, numerical examples are provided to support the theoretical analysis.     

Isoparametric finite-element approximation of a Steklov eigenvalue problem

We study the isoparametric variant of the finite-element method(FEM) for an approximation of Steklov eigenvalue problems forsecond-order, selfadjoint, elliptic differential operators.Error estimates for eigenfunctions and eigenvalues are derived.We prove the same estimate for eigenvalues as that obtainedin the case of conforming finite elements provided that theboundary of the domain is well approximated. Some algorithmicaspects arising from the FE isoparametric discretization ofthe Steklov problems are analysed. We finish this paper withnumerical results confirming the considered theory.     

On the correction of finite difference eigenvalue approximations for sturm-liouville problems with general boundary conditions

When finite difference and finite element methods are used to approximate continuous (differential) eigenvalue problems, the resulting algebraic eigenvalues only yield accurate estimates for the fundamental and first few harmonics. One way around this difficulty would be to estimate the error between the differential and algebraic eigenvalues by some independent procedure and then use it to correct the algebraic eigenvalues. Such an estimate has been derived by Paine, de Hoog and Anderssen for the Liouville normal form with Dirichlet boundary conditions. In this paper, we extend their result to the Liouville normal form with general boundary conditions.Dedicated to Germund Dahlquist on the occasion of his 60th birthday.     

Asymptotic Correction of More Sturm–Liouville Eigenvalue Estimates

The asymptotic correction technique of Paine, de Hoog and Anderssen can dramatically improve the accuracy of finite difference or finite element eigenvalues at negligible extra cost if closed form expressions are available for the errors in a simpler related problem. This paper gives closed form expressions for the errors in the eigenvalues of certain Sturm–Liouville problems obtained by various methods, thereby increasing the range of problems for which asymptotic correction can achieve maximum efficiency. It also investigates implementation of the method for more general problems.     

Root Vectors for Geometrically Simple Multiparameter Eigenvalues

A class of multiparameter eigenvalue problems involving (generally) non self-adjoint and unbounded operators is studied. Bases for lower order root subspaces, at geometrically simple eigenvalues of Fredholm type of arbitrary finite index, are computed in terms of the underlying multiparameter system.     

TWO-GRID DISCRETIZATION SCHEMES OF THE NONCONFORMING FEM FOR EIGENVALUE PROBLEMS

This paper extends the two-grid discretization scheme of the conforming finite elements proposed by Xu and Zhou （Math. Comput., 70 （2001）, pp.17-25） to the nonconforming finite elements for eigenvalue problems. In particular, two two-grid discretization schemes based on Rayleigh quotient technique are proposed. By using these new schemes, the solution of an eigenvalue problem on a fine mesh is reduced to that on a much coarser mesh together with the solution of a linear algebraic system on the fine mesh. The resulting solution still maintains an asymptotically optimal accuracy. Comparing with the two-grid discretization scheme of the conforming finite elements, the main advantages of our new schemes are twofold when the mesh size is small enough. First, the lower bounds of the exact eigenvalues in our two-grid discretization schemes can be obtained. Second, the first eigenvalue given by the new schemes has much better accuracy than that obtained by solving the eigenvalue problems on the fine mesh directly.     

Incomplete block-matrix factorization iterative methods for convection-diffusion problems

Standard Galerkin finite element methods or finite difference methods for singular perturbation problems lead to strongly unsymmetric matrices, which furthermore are in general notM-matrices. Accordingly, preconditioned iterative methods such as preconditioned (generalized) conjugate gradient methods, which have turned out to be very successful for symmetric and positive definite problems, can fail to converge or require an excessive number of iterations for singular perturbation problems.This is not so much due to the asymmetry, as it is to the fact that the spectrum can have both eigenvalues with positive and negative real parts, or eigenvalues with arbitrary small positive real parts and nonnegligible imaginary parts. This will be the case for a standard Galerkin method, unless the meshparameterh is chosen excessively small. There exist other discretization methods, however, for which the corresponding bilinear form is coercive, whence its finite element matrix has only eigenvalues with positive real parts; in fact, the real parts are positive uniformly in the singular perturbation parameter.In the present paper we examine the streamline diffusion finite element method in this respect. It is found that incomplete block-matrix factorization methods, both on classical form and on an inverse-free (vectorizable) form, coupled with a general least squares conjugate gradient method, can work exceptionally well on this type of problem. The number of iterations is sometimes significantly smaller than for the corresponding almost symmetric problem where the velocity field is close to zero or the singular perturbation parameter =1.The 2 ^nd author's research was sponsored by Control Data Corporation through its PACER fellowship program.The 3 ^rd author's research was supported by the Netherlands organization for scientific research (NWO).On leave from the Institute of Mathematics, Academy of Science, 1090 Sofia, P.O. Box 373, Bulgaria.     

Geršgorin type theorems for quaternionic matrices

This paper aims to set an account of the left eigenvalue problems for real quaternionic (finite) matrices. In particular, we will present the Geršgorin type theorems for the left (and right) eigenvalues of square quaternionic matrices. We shall conclude the paper with examples showing and summarizing some differences between complex matrices and quaternionic matrices and right and left eigenvalues of quaternionic matrices.     

Tensor eigenvalue complementarity problems

This paper studies tensor eigenvalue complementarity problems. Basic properties of standard and complementarity tensor eigenvalues are discussed. We formulate tensor eigenvalue complementarity problems as constrained polynomial optimization. When one tensor is strictly copositive, the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity. When no tensor is strictly copositive, we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process. The complementarity eigenvalues can be computed sequentially. The formulated polynomial optimization can be solved by Lasserre’s hierarchy of semidefinite relaxations. We show that it has finite convergence for generic tensors. Numerical experiments are presented to show the efficiency of proposed methods.     

Two-sided bounds of eigenvalues of second-and fourth-order elliptic operators

This article presents an idea in the finite element methods (FEMs) for obtaining two-sided bounds of exact eigenvalues. This approach is based on the combination of nonconforming methods giving lower bounds of the eigenvalues and a postprocessing technique using conforming finite elements. Our results hold for the second and fourth-order problems defined on two-dimensional domains. First, we list analytic and experimental results concerning triangular and rectangular nonconforming elements which give at least asymptotically lower bounds of the exact eigenvalues. We present some new numerical experiments for the plate bending problem on a rectangular domain. The main result is that if we know an estimate from below by nonconforming FEM, then by using a postprocessing procedure we can obtain two-sided bounds of the first (essential) eigenvalue. For the other eigenvalues λ_l, l = 2, 3, …, we prove and give conditions when this method is applicable. Finally, the numerical results presented and discussed in the paper illustrate the efficiency of our method.     

On the number of negative eigenvalues of a certain type of selfadjoint elliptic operators

This paper deals with several eigenvalue problems that have a finite number of negative eigenvalues. A connection is established between the number of negative eigenvalues, the parameters of the equation and some properties of the domain, through the correspondence of the eigenvalues with those of a related Stekloff problem.     

带谱参数边界条件的四阶边值问题的矩阵表示

研究了一类具有有限谱的带有谱参数边界条件的四阶微分方程边值问题及其矩阵表示,证明了对任意正整数m,所考虑的问题至多有2m+6个特征值,进一步给出这类带有谱参数边条件的四阶边值问题与一类矩阵特征值问题之间在具有相同特征值的意义下是等价的.     

Lower bounds for eigenvalues and postprocessing by an integral type nonconforming FEM

In this paper, we analyze some approximation properties of a nonconforming piecewise linear finite element with integral degrees of freedom. A nonconforming finite element method (FEM) is applied to second-order eigenvalue problems (EVPs).We prove that the eigenvalues computed by means of this element are smaller than the exact ones if the mesh size is small enough. The case when an EVP is defined on a nonconvex domain is considered. A superconvergent rate is established to a second-order elliptic problem by the introduction of nonstandard interpolated elements based on the integral type linear element. A simple postprocessing method applied to second-order EVPs is also proposed and analyzed. Finally, computational aspects are discussed and numerical examples are presented.     

On solution uniqueness of elliptic boundary value problems

In this paper, we consider the problem of solution uniqueness for the second order elliptic boundary value problem, by looking at its finite element or finite difference approximations. We derive several equivalent conditions, which are simpler and easier than the boundedness of the entries of the inverse matrix given in Yamamoto et al., [T. Yamamoto, S. Oishi, Q. Fang, Discretization principles for linear two-point boundary value problems, II, Numer. Funct. Anal. Optim. 29 (2008) 213–224]. The numerical experiments are provided to support the analysis made. Strictly speaking, the uniqueness of solution is equivalent to the existence of nonzero eigenvalues in the corresponding eigenvalue problem, and this condition should be checked by solving the corresponding eigenvalue problems. An application of the equivalent conditions is that we may discover the uniqueness simultaneously, while seeking the approximate solutions of elliptic boundary equations.     

Eigenvalue analysis of structures with interval parameters using the second-order Taylor series expansion and the DCA for QB

When the uncertainties in interval parameters are fairly large, the current analysis methods, which are usually based on the information of the first-order partial derivatives of eigenvalues, may not work well for the structural eigenvalue problem with interval parameters. To overcome this drawback, in this work, the structural eigenvalue problem with interval parameters is modeled as a series of QB (quadratic programming with box constrains) problems by taking advantage of the information of the second-order partial derivatives of eigenvalues. Then the series of QB problems would be solved by using the DCA (difference of convex functions algorithm) which is turn out be very effective for the QB problem. The specific examples, a concrete frame with sixty bars and a plate discretized with 300 finite elements, are given to show the effectiveness and feasibility of the proposed method compared with other methods.     

A finite element lumped mass scheme for solving eigenvalue problems of circular arches

Summary In this paper, we present a finite element lumped mass scheme for eigenvalue problems of circular arch structures, and give error estimates for the approximation. They assert that approximate eigenvalues and eigenfuctions converge to the exact ones. Some numerical examples are also given to illustrate our results.     

关于有限元离散方程特征值的界

各种有限元离散方程(包括协调元、非协调元及混合元等)的特征值的上、下界以及条件数的估计,早已引起了注意(见[1],[2],[5]).这里我们用一种简明的方法来处理这类问题.本文用到的关于Sobolev空间中的标准符号见[3].下面出现的常数c,c_1,c_2等在不同地方可能取不同的值.     

A new symmetric five-diagonal finite difference method for computing eigenvalues of fourth-order two-point boundary value problems

We present a new symmetric five-diagonal finite difference method for computing eigenvalues of two-point boundary value problems involving a fourth-order differential equation.     

The loss of stability of symmetrical equilibrium positions

Systems of differential equations possessing a finite (or compact) symmetry group and depending on one parameter are considered. The nature of the loss of stability of equilibrium positions is investigated in cases when, owing to symmetry, the linearized problem has multiple eigenvalues. Conditions are presented that determine whether the loss of stability when the parameter is varied is soft or hard, for double eigenvalues λ - zero or pure imaginary. Cases of triple zero eigenvalues λ corresponding to tetrahedral (or cubic) symmetry, are considered.     

Inverse mode problems for the finite element model of a vibrating rod

The inverse mode problems for the finite element model of an axially vibrating rod are formulated and solved. It is known that for the finite element model, based on linear shape functions, of the rod, the mass and stiffness matrices are both tridiagonal. It is shown that the finite element model of the rod can be constructed from two eigenvalues, their corresponding eigenvectors and the total mass of the rod. The necessary and sufficient conditions for the construction of a physically realizable rod with positive mass and stiffness elements from two eigenpairs and the total mass of the rod are established. If these conditions are satisfied, then the construction of the model is unique.