An accurate a posteriori error estimator for the Steklov eigenvalue problem and its applications

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine mesh and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.

     

An a posteriori error estimator based on shifts for positive Hermitian eigenvalue problems

This work deals with an a posteriori error estimator for Hermitian positive eigenvalue problems. The proposed estimator is based on the residual and the definition of suitable shifts in the matrix spectrum. The mathematical properties (certification and sharpness) are investigated and some numerical experiments are proposed.     

A posteriori error estimates for a discontinuous Galerkin approximation of Steklov eigenvalue problems

We derive a residual-based a posteriori error estimator for a discontinuous Galerkin approximation of the Steklov eigenvalue problem. Moreover, we prove the reliability and efficiency of the error estimator. Numerical results are provided to verify our theoretical findings.     

An a posteriori error estimator for anisotropic refinement

Summary. Besides an algorithm for local refinement, an a posteriori error estimator is the basic tool of every adaptive finite element method. Using information generated by such an error estimator the refinement of the grid is controlled. For 2nd order elliptic problems we present an error estimator for anisotropically refined grids, like -d cuboidal and 3-d prismatic grids, that gives correct information about the size of the error; additionally it generates information about the direction into which some element has to be refined to reduce the error in a proper way. Numerical examples are presented for 2-d rectangular and 3-d prismatic grids. Received March 15, 1994 / Revised version received June 3, 1994     

An a posteriori error estimator for plane-strain geotechnical analyses

In this paper, an a posteriori error estimator suitable for use in geotechnical engineering has been developed. This error criterion is based on the superconvergent patch recovery method with equilibrium and boundary conditions method, taking into account Biot's coupled consolidation theory. After testing on several problems, the introduction of pore pressures and other parameters into the error estimator was found to improve results significantly, providing a convergence rate of approximately two orders higher than the global rate.     

Local a priori/a posteriori error estimates of conforming finite elements approximation for Steklov eigenvalue problems

Based on the work of Xu and Zhou(2000),this paper makes a further discussion on conforming finite elements approximation for Steklov eigenvalue problems,and proves a local a priori error estimate and a new local a posteriori error estimate in ||·||1,Ω0 norm for conforming elements eigenfunction,which has not been studied in existing literatures.     

A posteriori error estimator for eigenvalue problems by mixed finite element method

In this paper,a residual type of a posteriori error estimator for the general second order elliptic eigenpair approximation by the mixed finite element method is derived and analyzed,based on a type of superconvergence result of the eigenfunction approximation.Its efficiency and reliability are proved by both theoretical analysis and numerical experiments.     

Steklov特征值问题的一种有效的Legendre-Galerkin谱逼近

本文给出Steklov特征值问题基于Legendre-Galerkin逼近的一种有效的谱方法.首先利用Legendre多项式构造了一组适当的基函数使得离散变分形式中的矩阵是稀疏的,然后推导了2维及3维情形下离散变分形式基于张量积的矩阵形式,由此可以快速地计算出离散的特征值和特征向量.文章还给出了误差分析和数值试验,数值结果表明本文提出的方法是稳定和有效的.     

Spectral Galerkin approximation and rigorous error analysis for the Steklov eigenvalue problem in circular domain

In this paper, we present spectral Galerkin approximation and rigorous error analysis for the Steklov eigenvalue problem in a circular domain. First of all, we use the polar coordinate transformation and technique of separation of variables to reduce the problem to a sequence of equivalent 1‐dimensional eigenvalue problems that can be solved individually in parallel. Then, we derive the pole conditions and introduce weighted Sobolev space according to pole conditions. Together with the approximate properties of orthogonal polynomials, we prove the error estimates of approximate eigenvalues for each 1‐dimensional eigenvalue problem. Finally, we provide some numerical experiments to validate the theoretical results and algorithms.     

An a posteriori error estimator for the Lame equation based on equilibrated fluxes

Katharina Witowski ^{We derive a new a posteriori error estimator for the Lamésystem based on H(div)-conforming elements and equilibratedfluxes. It is shown that the estimator gives rise to an upperbound where the constant is one up to higher-order terms. Thelower bound is also established using Argyris elements. Thereliability and efficiency of the proposed estimator are confirmedby some numerical tests.}     

A two-grid discretization scheme for the Steklov eigenvalue problem

In the paper, a two-grid discretization scheme is discussed for the Steklov eigenvalue problem. With the scheme, the solution of the Steklov eigenvalue problem on a fine grid is reduced to the solution of the Steklov eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid. Using spectral approximation theory, it is shown theoretically that the two-scale scheme is efficient and the approximate solution obtained by the scheme maintains the asymptotically optimal accuracy. Finally, numerical experiments are carried out to confirm the considered theory.     

Superconvergence and a posteriori error estimates for the Stokes eigenvalue problems

In this paper we consider the finite element approximation of the Stokes eigenvalue problems based on projection method, and derive some superconvergence results and the related recovery type a posteriori error estimators. The projection method is a postprocessing procedure that constructs a new approximation by using the least squares strategy. The results are based on some regularity assumptions for the Stokes equations, and are applicable to the finite element approximations of the Stokes eigenvalue problems with general quasi-regular partitions. Numerical results are presented to verify the superconvergence results and the efficiency of the recovery type a posteriori error estimators.     

Asymptotically exact a posteriori error estimator for biquadratic elements

This paper addresses the finite element method with a posteriori error estimation for elements of degree p = 1 and p = 2. It gives the formulae for the error indicators and error estimators. Basic mathematical characterizations of the estimators are given and it is shown that the estimators for p = 1 and p = 2 have different structures. Numerical examples show the effectivity of the approach and the high quality of the estimator.     

A posteriori error analysis for eigenvalue problems

A new a posteriori error analysis for symmetric eigenvalue problems of the adaptive finite element method (AFEM) is presented. Based on an H¹ stable L² projection, we prove reliability of the edge contribution for P₁ finite element methods. Hence the AFEM proposed can ignore the volume contribution as well as oscillations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Extrapolation and superconvergence of the Steklov eigenvalue problem

On the basis of a transform lemma, an asymptotic expansion of the bilinear finite element is derived over graded meshes for the Steklov eigenvalue problem, such that the Richardson extrapolation can be applied to increase the accuracy of the approximation, from which the approximation of O(h ^3.5) is obtained. In addition, by means of the Rayleigh quotient acceleration technique and an interpolation postprocessing method, the superconvergence of the bilinear finite element is presented over graded meshes for the Steklov eigenvalue problem, and the approximation of O(h ³) is gained. Finally, numerical experiments are provided to demonstrate the theoretical results.     

A residual based a posteriori estimator for the reaction-diffusion problem

A residual based a posteriori estimator for the reaction-diffusion problem is introduced. We show that the estimator gives both an upper and a lower bound to error. Numerical results are presented. To cite this article: M. Juntunen, R. Stenberg, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On the first eigenvalue of a fourth order Steklov problem

We prove some results about the first Steklov eigenvalue d ₁ of the biharmonic operator in bounded domains. Firstly, we show that Fichera’s principle of duality (Fichera in Atti Accad Naz Lincei 19:411–418, 1955) may be extended to a wide class of nonsmooth domains. Next, we study the optimization of d ₁ for varying domains: we disprove a long-standing conjecture, we show some new and unexpected features and we suggest some challenging problems. Finally, we prove several properties of the ball.