Error Estimation for Eigenvalues of Unbounded Linear Operators and an Application to Energy Levels in Graphene Quantum Dots

The eigenvalue problem for linear differential operators is important since eigenvalues correspond to the possible energy levels of a physical system. It is also important to have good estimates of the error in the computed eigenvalues. In this work, we use spline interpolation to construct approximate eigenfunctions of a linear operator using the corresponding eigenvectors of a discretized approximation of the operator. We show that an error estimate for the approximate eigenvalues can be obtained by evaluating the residual for an approximate eigenpair. The interpolation scheme is selected in such a way that the residual can be evaluated analytically. To demonstrate that the method gives useful error bounds, we apply it to a problem originating from the study of graphene quantum dots where the goal was to investigate the change in the spectrum from incorporating electron–electron interactions in the potential.     

Eigenvalue approximation from below using non-conforming finite elements

This is a survey article about using non-conforming finite elements in solving eigenvalue problems of elliptic operators,with emphasis on obtaining lower bounds. In addition,this article also contains some new materials for eigenvalue approximations of the Laplace operator,which include:1) the proof of the fact that the non-conforming Crouzeix-Raviart element approximates eigenvalues associated with smooth eigenfunctions from below;2) the proof of the fact that the non-conforming EQ rot1 element approximates eigenvalues from below on polygonal domains that can be decomposed into rectangular elements;3) the explanation of the phenomena that numerical eigenvalues λ 1,h and λ 3,h of the non-conforming Q rot1 element approximate the true eigenvalues from below for the L-shaped domain. Finally,we list several unsolved problems.     

A posteriori error control for finite element approximations of elliptic eigenvalue problems

We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error representations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process.     

Numerical computation of eigenvalues of discontinuous Sturm–Liouville problems with parameter dependent boundary conditions using sinc method

In this paper, we consider a Sturm–Liouville problem which contains an eigenparameter appearing linearly in two boundary conditions, in addition to an internal point of discontinuity. Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. We apply the sinc method, which is based on the sampling theory to compute approximations of the eigenvalues. An error analysis is exhibited involving rigorous error bounds. Using computable error bounds we obtain eigenvalue enclosures in a simple way. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

The combined effect of numerical integration and approximation of the boundary in the finite element method for eigenvalue problems

Summary. The paper deals with the finite element analysis of second order elliptic eigenvalue problems when the approximate domains are not subdomains of the original domain and when at the same time numerical integration is used for computing the involved bilinear forms. The considerations are restricted to piecewise linear approximations. The optimum rate of convergence for approximate eigenvalues is obtained provided that a quadrature formula of first degree of precision is used. In the case of a simple exact eigenvalue the optimum rate of convergence for approximate eigenfunctions in the -norm is proved while in the -norm an almost optimum rate of convergence (i.e. near to is achieved. In both cases a quadrature formula of first degree of precision is used. Quadrature formulas with degree of precision equal to zero are also analyzed and in the case when the exact eigenfunctions belong only to the convergence without the rate of convergence is proved. In the case of a multiple exact eigenvalue the approximate eigenfunctions are compard (in contrast to standard considerations) with linear combinations of exact eigenfunctions with coefficients not depending on the mesh parameter . Received September 18, 1993 / Revised version received September 26, 1994     

A New Implicit Iteration Process with Errors for the Family of Strongly Pseudocontractive Mappings

A method is presented for generating a sequence of lower and upper bounds for the eigenvalues of the problem (i) Tu-λSu = 0, where T and S belong to a class of unbounded and nonsymmetric operators in a separable Hilbert space. Sufficient conditions are derived for the convergence of the sequence of bounds to the eigenvalues of (i), and the applicability of the method is illustrated by approximating the smallest eigenvalue of a non-selfadjoint differential eigenvalue problem.     

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.     

Estimation of the effect of numerical integration in finite element eigenvalue approximation

Summary Finite element approximations of the eigenpairs of differential operators are computed as eigenpairs of matrices whose elements involve integrals which must be evaluated by numerical integration. The effect of this numerical integration on the eigenvalue and eigenfunction error is estimated. Specifically, for 2nd order selfadjoint eigenvalue problems we show that finite element approximations with quadrature satisfy the well-known estimates for approximations without quadrature, provided the quadrature rules have appropriate degrees of precision.The work of this author was partially supported by the National Science Foundation under Grant DMS-84-10324     

An integral method for solving nonlinear eigenvalue problems

We propose a numerical method for computing all eigenvalues (and the corresponding eigenvectors) of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane. The method uses complex integrals of the resolvent operator, applied to at least k column vectors, where k is the number of eigenvalues inside the contour. The theorem of Keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension k. No initial approximations of eigenvalues and eigenvectors are needed. The method is particularly suitable for moderately large eigenvalue problems where k is much smaller than the matrix dimension. We also give an extension of the method to the case where k is larger than the matrix dimension. The quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours. Using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour.     

Adaptive finite element methods for computing band gaps in photonic crystals

In this paper we propose and analyse adaptive finite element methods for computing the band structure of 2D periodic photonic crystals. The problem can be reduced to the computation of the discrete spectra of each member of a family of periodic Hermitian eigenvalue problems on a unit cell, parametrised by a two-dimensional parameter - the quasimomentum. These eigenvalue problems involve non-coercive elliptic operators with generally discontinuous coefficients and are solved by adaptive finite elements. We propose an error estimator of residual type and show it is reliable and efficient for each eigenvalue problem in the family. In particular we prove that if the error estimator converges to zero then the distance of the computed eigenfunction from the true eigenspace also converges to zero and the computed eigenvalue converges to a true eigenvalue with double the rate. We also prove that if the distance of a computed sequence of approximate eigenfunctions from the true eigenspace approaches zero, then so must the error estimator. The results hold for eigenvalues of any multiplicity. We illustrate the benefits of the resulting adaptive method in practice, both for fully periodic structures and also for the computation of eigenvalues in the band gap of structures with defect, using the supercell method.     

On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form

In the approximation of linear elliptic operators in mixed form, it is well known that the so-called inf-sup and ellipticity in the kernel properties are sufficient (and, in a sense to be made precise, necessary) in order to have good approximation properties and optimal error bounds. One might think, in the spirit of Mercier-Osborn-Rappaz-Raviart and in consideration of the good behavior of commonly used mixed elements (like Raviart-Thomas or Brezzi-Douglas-Marini elements), that these conditions are also sufficient to ensure good convergence properties for eigenvalues. In this paper we show that this is not the case. In particular we present examples of mixed finite element approximations that satisfy the above properties but exhibit spurious eigenvalues. Such bad behavior is proved analytically and demonstrated in numerical experiments. We also present additional assumptions (fulfilled by the commonly used mixed methods already mentioned) which guarantee optimal error bounds for eigenvalue approximations as well.

     

Higher order convergence results for the Rayleigh-Ritz method applied to eigenvalue problems: 2. Improved error bounds for eigenfunctions

The application of the Rayleigh-Ritz method for approximating the eigenvalues and eigenfunctions of linear eigenvalue problems in several dimensions is investigated. The object is to improve upon known error estimates for the approximate eigenfunctions. Results for the Galerkin approximation of the eigenfunctions are developed under varying assumptions on the boundary conditions and domain of definition of the eigenvalue problem. These results, coupled with a previous result relating Galerkin and Rayleigh-Ritz approximation of the eigenfunctions, are then used to obtain improved error estimates for the approximate eigenfunctions in theL ² and uniform norms.This research was supported in part by AEC Grant (11-1)-2075.     

Guaranteed Error Bounds for Eigenvalues of Singular Sturm‐Liouville Problems

We develop a simple oscillation theory for singular Sturm‐Liouville problems and combine it with recent asymptotic results, and with the AWA interval‐arithmetic code for integration of initial value problems with guaranteed error bounds, to obtain eigenvalue approximations with guaranteed error bounds for a class of singular Sturm‐Liouville problems. We believe that this is the first time that this has been achieved for singular eigenvalue problems.     

Verified numerical computations for multiple and nearly multiple eigenvalues of elliptic operators

In this paper, we propose a numerical method to verify bounds for multiple eigenvalues for elliptic eigenvalue problems. We calculate error bounds for approximations of multiple eigenvalues and base functions of the corresponding invariant subspaces. For matrix eigenvalue problems, Rump (Linear Algebra Appl. 324 (2001) 209) recently proposed a validated numerical method to compute multiple eigenvalues. In this paper, we extend his formulation to elliptic eigenvalue problems, combining it with a method developed by one of the authors (Jpn. J. Indust. Appl. Math. 16 (1998) 307).     

Realistic error bounds for a simple eigenvalue and its associated eigenvector

Summary An algorithm is described which, given an approximate simple eigenvalue and a corresponding approximate eigenvector, provides rigorous error bounds for improved versions of them. No information is required on the rest of the eigenvalues, which may indeed correspond to non-linear elementary divisors. A second algorithm is described which gives more accurate improved versions than the first but provides only error estimates rather than rigorous bounds. Both algorithms extend immediately to the generalized eigenvalue problem.Dedicated to A.S. Householder on his 75th birthday     

Perturbed preconditioned inverse iteration for operator eigenvalue problems with applications to adaptive wavelet discretization

In this paper we discuss an abstract iteration scheme for the calculation of the smallest eigenvalue of an elliptic operator eigenvalue problem. A short and geometric proof based on the preconditioned inverse iteration (PINVIT) for matrices (Knyazev and Neymeyr, SIAM J Matrix Anal 31:621–628, 2009) is extended to the case of operators. We show that convergence is retained up to any tolerance if one only uses approximate applications of operators which leads to the perturbed preconditioned inverse iteration (PPINVIT). We then analyze the Besov regularity of the eigenfunctions of the Poisson eigenvalue problem on a polygonal domain, showing the advantage of an adaptive solver to uniform refinement when using a stable wavelet base. A numerical example for PPINVIT, applied to the model problem on the L-shaped domain, is shown to reproduce the predicted behaviour.     

An a posteriori error estimation of the Rayleigh-Ritz eigenvalues

Summary LetA, B be essentially self-adjoint and positive definite differential operators defined inL ₂(G). Using Svirskij's construction of the base operator and some results from the analytic perturbation theory of linear operators a formula providing eigenvalue lower bounds of the problemAu=Bu is derived. In this formula a rough lower bound of some higher eigenvalue and the residual convergence of the Rayleigh-Ritz eigenfunction approximations are needed. Some numerical results are presented.     

A-posteriori error analysis and adaptive finite element methods for singularly perturbed convection-diffusion equations

For singularly perturbed one-dimensional convection-diffusion equations, finite element approximations are constructed based on a so-called approximate symmetrization of the given unsymmetric problem. Local a-posteriori error estimates are established with respect to an appropriate energy norm where the bounds are proved to be realistic. The local bounds, called error indicators, provide a basis for a self-adaptive mesh refinement. For a model problem numerical results are presented showing that the adaptive method detects and resolves the boundary layer.     

Convergence of the control parameterization Ritz method for nonlinear optimal control problems

This paper studies the local convergence properties of the control parameterization Ritz method in which the control variable is approximated over a finite-dimensional subspace. The nonlinear free-endpoint optimal control problem is considered, and error bounds are derived for both the cost functional and state-control convergence. Explicit error bounds are obtained for the particular case of approximations over spline spaces. On specializing the general results to the linear-quadratic regulator problem, global convergence results are obtained. Computational results supporting the theoretically derived error bounds are presented.This research was supported by the University Grants Committee of New Zealand.     

Enclosing clusters of zeros of polynomials

Lagrange interpolation and partial fraction expansion can be used to derive a Gerschgorin-type theorem that gives simple and powerful a posteriori error bounds for the zeros of a polynomial if approximations to all zeros are available. Compared to bounds from a corresponding eigenvalue problem, a factor of at least two is gained.The accuracy of the bounds is analyzed, and special attention is given to ensure that the bounds work well not only for single zeros but also for multiple zeros and clusters of close zeros.A Rouché-type theorem is also given, that in many cases reduces the bound even further.