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.     

A Two-Level Method for Nonsymmetric Eigenvalue Problems

A two-level discretization method for eigenvalue problems is studied. Compared to the standard Galerkin finite element discretization technique performed on a fine grid this method discretizes the eigenvalue problem on a coarse grid and obtains an improved eigenvector (eigenvalue) approximation by solving only a linear problem on the fine grid (or two linear problems for the case of eigenvalue approximation of nonsymmetric problems). The improved solution has the asymptotic accuracy of the Galerkin discretization solution. The link between the method and the iterated Galerkin method is established. Error estimates for the general nonsymmetric case are derived.     

Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations

The paper deals with error estimates and lower bound approximations of the Steklov eigenvalue problems on convex or concave domains by nonconforming finite element methods. We consider four types of nonconforming finite elements: Crouzeix-Raviart, Q ₁ ^rot , EQ ₁ ^rot and enriched Crouzeix-Raviart. We first derive error estimates for the nonconforming finite element approximations of the Steklov eigenvalue problem and then give the analysis of lower bound approximations. Some numerical results are presented to validate our theoretical results.     

A note on the residual type a posteriori error estimates for finite element eigenpairs of nonsymmetric elliptic eigenvalue problems

In this paper we study the residual type a posteriori error estimates for general elliptic (not necessarily symmetric) eigenvalue problems. We present estimates for approximations of semisimple eigenvalues and associated eigenvectors. In particular, we obtain the following new results: 1) An error representation formula which we use to reduce the analysis of the eigenvalue problem to the analysis of the associated source problem; 2) A local lower bound for the error of an approximate finite element eigenfunction in a neighborhood of a given mesh element T.     

Extrapolation of Mixed Finite Element Approximations for the Maxwell Eigenvalue Problem

In this paper,a general method to derive asymptotic error expansion formulas for the mixed finite element approximations of the Maxwell eigenvalue problem is established.Abstract lemmas for the error of the eigenvalue approximations are obtained.Based on the asymptotic error expansion formulas,the Richardson extrapolation method is employed to improve the accuracy of the approximations for the eigenvalues of the Maxwell system from θ(h2) to θ(h4) when applying the lowest order Nédé1ec mixed finite element and a nonconforming mixed finite element.To our best knowledge,this is the first superconvergence result of the Maxwell eigenvalue problem by the extrapolation of the mixed finite element approximation.Numerical experiments are provided to demonstrate the theoretical results.     

A Multilevel Correction Method for Steklov Eigenvalue Problem by Nonconforming Finite Element Methods

In this paper, a multilevel correction scheme is proposed to solve the Steklov eigenvalue problem by nonconforming finite element methods. With this new scheme, the accuracy of eigenpair approximations can be improved after each correction step which only needs to solve a source problem on finer finite element space and an Steklov eigenvalue problem on the coarsest finite element space. This correction scheme can increase the overall efficiency of solving eigenvalue problems by the nonconforming finite element method. Furthermore, as same as the direct eigenvalue solving by nonconforming finite element methods, this multilevel correction method can also produce the lower-bound approximations of the eigenvalues.     

特征值问题的三角形二次元的展开式及其外推

我们考虑利用三角形二次元来求解特征值问题,并给出特征值的误差展开式,以此为基础进行外推获得高精度.     

Backward errors for eigenvalues and eigenvectors of Hermitian,skew-Hermitian,H-even and H-odd matrix polynomials

We discuss the perturbation analysis for eigenvalues and eigenvectors of structured homogeneous matrix polynomials with Hermitian, skew-Hermitian, H-even and H-odd structure. We construct minimal structured perturbations (structured backward errors) such that an approximate eigenvalue and eigenvector pair (finite or infinite eigenvalues) is an exact eigenvalue eigenvector pair of an appropriately perturbed structured matrix polynomial. We present various comparisons with unstructured backward errors and previous backward errors constructed for the non-homogeneous case and show that our results generalize previous results.     

Recovery type a posteriori estimates and superconvergence for nonconforming FEM of eigenvalue problems

The main goal of this paper is to present recovery type a posteriori error estimators and superconvergence for the nonconforming finite element eigenvalue approximation of self-adjoint elliptic equations by projection methods. Based on the superconvergence results of nonconforming finite element for the eigenfunction we derive superconvergence and recovery type a posteriori error estimates of the eigenvalue. The results are based on some regularity assumption for the elliptic problem and are applicable to the lowest order nonconforming finite element approximations of self-adjoint elliptic eigenvalue problems with quasi-regular partitions. Therefore, the results of this paper can be employed to provide useful a posteriori error estimators in practical computing under unstructured meshes.     

Two-scale sparse finite element approximations

To reduce computational cost,we study some two-scale finite element approximations on sparse grids for elliptic partial differential equations of second order in a general setting.Over any tensor product domain ?R~d with d = 2,3,we construct the two-scale finite element approximations for both boundary value and eigenvalue problems by using a Boolean sum of some existing finite element approximations on a coarse grid and some univariate fine grids and hence they are cheaper approximations.As applications,we obtain some new efficient finite element discretizations for the two classes of problem:The new two-scale finite element approximation on a sparse grid not only has the less degrees of freedom but also achieves a good accuracy of approximation.     

Asymptotic expansions andL ∞-error estimates for mixed finite element methods for second order elliptic problems

Summary Asymptotic expansions for mixed finite element approximations of the second order elliptic problem are derived and Richardson extrapolation can be applied to increase the accuracy of the approximations. A new procedure, which is called the error corrected method, is presented as a further application of the asymptotic error expansion for the first order BDM approximation of the scalar field. The key point in deriving the asymptotic expansions for the error is an establishment ofL ¹-error estimates for mixed finite element approximations for the regularized Green's functions. As another application of theL ¹-error estimates for the regularized Green's functions, we shall present maximum norm error estimates for mixed finite element methods for second order elliptic problems.     

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     

On approximating the solution of the non-stationary Stokes equations using the cell discretization algorithm

The cell discretization algorithm, a nonconforming extension of the finite element method, is used to obtain approximations to the velocity and pressure satisfying the nonstationary Stokes equations. Error estimates show convergence of the approximations. An implementation using polynomial bases is described that permits the use of the continuous approximations of the h–p finite element method and exactly satisfies the solenoidal requirement. We express the error estimates in terms of the diameter h of a cell and the degree p of the approximation on each cell. Results of an experiment with p10 are presented that confirm the theoretical estimates.     

Error estimates using the cell discretization method for second-order hyperbolic equations

The cell discretization algorithm provides approximate solutions to second-order hyperbolic equations with coefficients independent of time. We obtain error estimates that show general convergence for homogeneous problems using semi-discrete approximations. A polynomial implementation of the algorithm is described that is a nonconforming extension of the finite element method that can also produce the continuous approximations of an h–p finite element method. Numerical tests are made that confirm the theoretical estimates. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 531–548, 1997     

Adaptive computation of smallest eigenvalues of self‐adjoint elliptic partial differential equations

We consider a new adaptive finite element (AFEM) algorithm for self‐adjoint elliptic PDE eigenvalue problems. In contrast to other approaches we incorporate the inexact solutions of the resulting finite‐dimensional algebraic eigenvalue problems into the adaptation process. In this way we can balance the costs of the adaptive refinement of the mesh with the costs for the iterative eigenvalue method. We present error estimates that incorporate the discretization errors, approximation errors in the eigenvalue solver and roundoff errors, and use these for the adaptation process. We show that it is also possible to restrict to very few iterations of a Krylov subspace solver for the eigenvalue problem on coarse meshes. Several examples are presented to show that this new approach achieves much better complexity than the previous AFEM approaches which assume that the algebraic eigenvalue problem is solved to full accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

On solenoidal high‐degree polynomial approximations to solutions of the stationary Stokes equations

The cell discretization algorithm, a nonconforming extension of the finite element method, is used to obtain approximations to the velocity and pressure functions satisfying the Stokes equations. Error estimates show convergence of the method. An implementation using polynomial bases is described that permits the use of the continuous approximations of the h‐p finite element method and exactly satisfies the solenoidal requirement. We express the error estimates in terms of the diameter h of a cell and degree p of the approximation on each cell. Examples of 10^th degree polynomial approximations are described that substantiate the theoretical estimates. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 480–493, 2000     

A Posteriori Error Estimates for Distributed Convex Optimal Control Problems

In this paper, we present an a posteriori error analysis for finite element approximation of distributed convex optimal control problems. We derive a posteriori error estimates for the coupled state and control approximations under some assumptions which hold in many applications. Such estimates, which are apparently not available in the literature, can be used to construct reliable adaptive finite element approximation schemes for control problems. Explicit estimates are obtained for some model problems which frequently appear in real-life applications.     

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.     

Multiscale finite element discretizations based on local defect correction for the biharmonic eigenvalue problem of plate buckling

In this paper, we study the multiscale finite element discretizations about the biharmonic eigenvalue problem of plate buckling. On the basis of the work of Dai and Zhou (SIAM J. Numer. Anal. 46[1] [2008] 295‐324), we establish a three‐scale scheme, a multiscale discretization scheme, and the associated parallel version based on local defect correction. We first prove a local priori error estimate of finite element approximations, then give the error estimates of multiscale discretization schemes. Theoretical analysis and numerical experiments indicate that our schemes are suitable and efficient for eigenfunctions with local low smoothness.     

Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods

In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, Q ₁^rot and EQ ₁^rot. Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. This project is supported in part by the National Natural Science Foundation of China (10471103) and is subsidized by the National Basic Research Program of China under the grant 2005CB321701.