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.     

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.     

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.     

Cluster robust error estimates for the Rayleigh-Ritz approximation II: Estimates for eigenvalues

This is the second part of a paper that deals with error estimates for the Rayleigh-Ritz approximations of the spectrum and invariant subspaces of a bounded Hermitian operator in a Hilbert or Euclidean space. This part addresses the approximation of eigenvalues. Two kinds of estimates are considered: (i) estimates for the eigenvalue errors via the best approximation errors for the corresponding invariant subspaces, and (ii) estimates for the same via the corresponding residuals. Estimates of these two kinds are needed for, respectively, the a priori and a posteriory error analysis of numerical methods for computing eigenvalues. The paper’s major concern is to ensure that the estimates in question are accurate and ‘cluster robust’, i.e. are not adversely affected by the presence of clustered, i.e. closely situated eigenvalues among those of interest. The paper’s main new results introduce estimates for clustered eigenvalues whereby not only the distances between eigenvalues in the cluster are not present but also the distances between the cluster and the rest of the spectrum appear in asymptotically insignificant terms only.     

非对称特征值问题的有限元渐进准确后验误差估计

本文致力于二阶椭圆非对称特征值问题的基于保多项式重构(PPR)的后验误差分析.我们提出了一个用于求解对流占有的非对称特征值问题的渐近精确后验误差估计子(特别是针对非光滑特征函数或多个特征值的情况).数值例子验证了我们的理论分析.     

Adaptive finite element algorithms for eigenvalue problems based on local averaging type a posteriori error estimates

The local averaging technique has become a popular tool in adaptive finite element methods for solving partial differential boundary value problems since it provides efficient a posteriori error estimates by a simple postprocessing. In this paper, the technique is introduced to solve a class of symmetric eigenvalue problems. Its efficiency and reliability are proved by both the theory and numerical experiments structured meshes as well as irregular meshes. Dedicated to Charles A. Micchelli on his 60th birthday Mathematics subject classifications (2000) 65N15, 65N25, 65N30, 65N50. Subsidized by the Special Funds for Major State Basic Research Projects, and also supported in part by the Chinese National Natural Science Foundation and the Knowledge Innovation Program of the Chinese Academy of Sciences.     

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.     

本征值Wilson非协调元近似的超收敛性与后验误差估计

本文给出二阶椭圆本征值问题Ｗｉｌｓｏｎ非协调元的超收敛与后验误差估计式,花很少代价就把Ｗｉｌｓｏｎ近似本征值的精度阶从ｈ＾２提高到ｈ＾４,并得到了渐近准确误差指示子。