A multilevel correction method for Stokes eigenvalue problems and its applications

In this paper, a new multilevel correction scheme is proposed to solve Stokes eigenvalue problems by the finite element method. This new scheme contains a series of correction steps, and the accuracy of eigenpair approximation can be improved after each step. In each correction step, we only need to solve a Stokes problem on the corresponding fine finite element space and a Stokes eigenvalue problem on the coarsest finite element space. This correction scheme can improve the efficiency of solving Stokes eigenvalue problems by the finite element method. As applications of this multilevel correction method, a multigrid method and an adaptive finite element technique are introduced for Stokes eigenvalue problems. Some numerical results are given to validate our schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

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.

     

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

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 meshes 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.     

A new type of full multigrid method for the elasticity eigenvalue problem

This paper introduces a new type of full multigrid method for the elasticity eigenvalue problem. The main idea is to avoid solving large scale elasticity eigenvalue problem directly by transforming the solution of the elasticity eigenvalue problem into a series of solutions of linear boundary value problems defined on a multilevel finite element space sequence and some small scale elasticity eigenvalue problems defined on the coarsest correction space. The involved linear boundary value problems will be solved by performing some multigrid iterations. Besides, some efficient techniques such as parallel computing and adaptive mesh refinement can also be absorbed in our algorithm. The efficiency and validity of the multigrid methods are verified by several numerical experiments.     

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.     

Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem

In this paper, we propose a method to improve the convergence rate of the lowest order Raviart-Thomas mixed finite element approximations for the second order elliptic eigenvalue problem. Here, we prove a supercloseness result for the eigenfunction approximations and use a type of finite element postprocessing operator to construct an auxiliary source problem. Then solving the auxiliary additional source problem on an augmented mixed finite element space constructed by refining the mesh or by using the same mesh but increasing the order of corresponding mixed finite element space, we can increase the convergence order of the eigenpair approximation. This postprocessing method costs less computation than solving the eigenvalue problem on the finer mesh directly. Some numerical results are used to confirm the theoretical analysis.     

A full multigrid method for nonlinear eigenvalue problems

We introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of the nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We prove that the computational work of this new scheme is truly optimal, the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.     

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.     

Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems

In this paper we propose a method for improving the convergence rate of the mixed finite element approximations for the Stokes eigenvalue problem. It is based on a postprocessing strategy that consists of solving an additional Stokes source problem on an augmented mixed finite element space which can be constructed either by refining the mesh or by using the same mesh but increasing the order of the mixed finite element space. Dedicated to Ivan Hlaváček on the occasion of his 75th birthday     

Multilevel preconditioners for discretizations of the biharmonic equation by rectangular finite elements

Recently, some new multilevel preconditioners for solving elliptic finite element discretizations by iterative methods have been proposed. They are based on appropriate splittings of the finite element spaces under consideration, and may be analyzed within the framework of additive Schwarz schemes. In this paper we discuss some multilevel methods for discretizations of the fourth-order biharmonic problem by rectangular elements and derive optimal estimates for the condition numbers of the preconditioned linear systems. For the Bogner–Fox–Schmit rectangle, the generalization of the Bramble–Pasciak–Xu method is discussed. As a byproduct, an optimal multilevel preconditioner for nonconforming discretizations by Adini elements is also derived.     

Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods

We introduce some ways to compute the lower and upper bounds of the Laplace eigenvalue problem.By using the special nonconforming finite elements,i.e.,enriched Crouzeix-Raviart element and extended Q1ro t,we get the lower bound of the eigenvalue.Additionally,we use conforming finite elements to do the postprocessing to get the upper bound of the eigenvalue,which only needs to solve the corresponding source problems and a small eigenvalue problem if higher order postprocessing method is implemented.Thus,we can obtain the lower and upper bounds of the eigenvalues simultaneously by solving eigenvalue problem only once.Some numerical results are also presented to demonstrate our theoretical analysis.     

Application of nonconforming finite elements to solving advection—diffusion problems

The paper investigates some nonconforming finite elements and nonconforming finite element schemes for solving an advection—diffusion equation. This investigation is aimed at finding new schemes for solving parabolic equations. The study uses a finite element method, variational-difference schemes, and test calculations. Two types of schemes are examined: one is obtained with the help of the Bubnov—Galerkin method from a weak problem determination (nonmonotone scheme), and the other one is a monotone up-stream scheme obtained from an approximate weak problem determination with a special approximation of the skew-symmetric terms.     

Local and parallel finite element method for solving the biharmonic eigenvalue problem of plate vibration

In this paper, we establish a new local and parallel finite element discrete scheme based on the shifted‐inverse power method for solving the biharmonic eigenvalue problem of plate vibration. We prove the local error estimation of finite element solution for the biharmonic equation/eigenvalue problem and prove the error estimation of approximate solution obtained by the local and parallel scheme. When the diameters of three grids satisfy H⁴ = ?(w²) = ?(h), the approximate solutions obtained by our schemes can achieve the asymptotically optimal accuracy. The numerical experiments show that the computational schemes proposed in this paper are effective to solve the biharmonic eigenvalue problem of plate vibration.     

Numerical analysis of a non‐singular boundary integral method: Part I. The circular case

In order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we present here a method which consists in inverting, on a finite element space, a non‐singular integral operator. This operator is a geometrical perturbation of the Steklov operator, and we precisely define the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme. Furthermore, this numerical scheme does not give rise to any singular integral. The scheme can also be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single layer potential, the special quadrature formula being defined by the introduction of a neighbouring curve. In the present paper, we prove stability and we give error estimates of our numerical scheme when the Laplace problem is set on a disk. We will extend our results to any domains by using compact perturbation arguments, in a second paper. Copyright © 2001 John Wiley & Sons, Ltd.     

On the Discrete Poincaré–Friedrichs Inequalities for Nonconforming Approximations of the Sobolev Space H 1

We present a direct proof of the discrete Poincaré–Friedrichs inequalities for a class of nonconforming approximations of the Sobolev space H ¹(Ω), indicate optimal values of the constants in these inequalities, and extend the discrete Friedrichs inequality onto domains only bounded in one direction. We consider a polygonal domain Ω in two or three space dimensions and its shape-regular simplicial triangulation. The nonconforming approximations of H ¹(Ω) consist of functions from H ¹ on each element such that the mean values of their traces on interelement boundaries coincide. The key idea is to extend the proof of the discrete Poincaré–Friedrichs inequalities for piecewise constant functions used in the finite volume method. The results have applications in the analysis of nonconforming numerical methods, such as nonconforming finite element or discontinuous Galerkin methods.     

An immersed finite element space and its approximation capability

This article discusses an immersed finite element (IFE) space introduced for solving a second‐order elliptic boundary value problem with discontinuous coefficients (interface problem). The IFE space is nonconforming and its partition can be independent of the interface. The error estimates for the interpolation of a function in the usual Sobolev space indicate that this IFE space has an approximation capability similar to that of the standard conforming linear finite element space based on body‐fit partitions. Numerical examples of the related finite element method based on this IFE space are provided. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 338–367, 2004     

Order Reduced Schemes for the Fourth Order Eigenvalue Problems on Multi-Connected Planar Domains

In this paper, we study the order reduced finite element method for the fourth order eigenvalue problems on multi-connected planar domains. Particularly, we take the biharmonic and the Helmholtz transmission eigenvalue problems as model problems, present for each an equivalent order reduced formulation and a corresponding stable discretization scheme, and present rigorous theoretical analysis. The schemes are readily fit for multilevel correction algorithms with optimal computational costs. Numerical experiments are given for verifications.     

非定常对流扩散问题的非协调局部投影有限元方法

本文将近年来基于协调有限元逼近提出的涡旋粘性法推广应用到非协调有限元逼近,对非定常的对流占优扩散问题,空间采用非协调Crouzeix-Raviart元逼近,时间用Crank-Nicolson差分离散格式,提出了Crank-Nicolson差分-局部投影法稳定化有限元格式,我们对稳定性和误差估计给出了详细的分析,得出了最...     

Approximation capability of a bilinear immersed finite element space

This article discusses a bilinear immersed finite element (IFE) space for solving second‐order elliptic boundary value problems with discontinuous coefficients (interface problem). This is a nonconforming finite element space and its partition can be independent of the interface. The error estimates for the interpolation of a Sobolev function indicate that this IFE space has the usual approximation capability expected from bilinear polynomials. Numerical examples of the related finite element method are provided. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008