特征值问题的自适应反迭代有限元算法

1引言设Ω∈R~2为Lipschitz单连通的有界闭区域,X为定义在Ω的Sobolev空间,a(·,·)和b(·,·)为X×X→C的有界双线性或半双线性泛函,考虑变分特征值问题:求(λ,u≠0)∈C×X使得a(u,v)=λb(u,u),(?)u∈X,其中a(·,·)满足X上的"V-强制性"条件或者连续的inf-sup条件,设M_h为Q区域上的正则三角形剖分,X_h∈X为定义在M_h有限元子空间,上述变分问题对应的有限元离散问题为:求(λ_h,u_h)∈R×X,u_h≠0使得     

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.     

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.     

On Finite Element Methods for Inhomogeneous Dielectric Waveguides

We investigate the problem of computing electromagnetic guided waves in a closed,inhomogeneous, pillared three-dimensional waveguide at a given frequency. The problem is formulated as a generalized eigenvalue problem. By modifying the sesquilinear form associated with the eigenvalue problem, we provide a new convergence analysis for the finite element approximations. Numerical results are reported to illustrate the performance of the method.     

A new adaptive mixed finite element method based on residual type a posterior error estimates for the Stokes eigenvalue problem

In this article, we combine mixed finite element method, multiscale discretization, and Rayleigh quotient iteration to propose a new adaptive algorithm based on residual type a posterior error estimates for the Stokes eigenvalue problem. Both reliability and efficiency of the error indicator are proved. The efficiency of the algorithm is also investigated using Chen's Innovation Finite Element Method (iFEM) package. Numerical results are satisfying.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 31–53, 2015     

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.     

A posteriori error estimate for discontinuous Galerkin finite element method on polytopal mesh

In this work, we derive a posteriori error estimates for discontinuous Galerkin finite element method on polytopal mesh. We construct a reliable and efficient a posteriori error estimator on general polygonal or polyhedral meshes. An adaptive algorithm based on the error estimator and DG method is proposed to solve a variety of test problems. Numerical experiments are performed to illustrate the effectiveness of the algorithm.     

New adaptive mixed finite element method (AMFEM)

The need to develop reliable and efficient adaptive algorithms using mixed finite element methods arises from various applications in fluid dynamics and computational continuum mechanics. In order to save degrees of freedom, not all but just some selected set of finite element domains are refined and hence the fundamental question of convergence requires a new mathematical argument as well as the question of optimality. We will present a new adaptive algorithm for mixed finite element methods to solve the model Poisson problem, for which optimal convergence can be proved. The a posteriori error control of mixed finite element methods dates back to Alonso (1996) Error estimators for a mixed method. and Carstensen (1997) A posteriori error estimate for the mixed finite element method. The error reduction and convergence for adaptive mixed finite element methods has already been proven by Carstensen and Hoppe (2006) Error Reduction and Convergence for an Adaptive Mixed Finite Element Method, Convergence analysis of an adaptive nonconforming finite element methods. Recently, Chen, Holst and Xu (2008) Convergence and Optimality of Adaptive Mixed Finite Element Methods. presented convergence and optimality for adaptive mixed finite element methods following arguments of Rob Stevenson for the conforming finite element method. Their algorithm reduces oscillations, before applying and a standard adaptive algorithm based on usual error estimation. The proposed algorithm does this in a natural way, by switching between the reduction of either the estimated error or oscillations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the h-adaptive coupling of FE and BE for viscoplastic and elasto-plastic interface problems

This paper presents a posteriori error estimates for the symmetric finite element and boundary element coupling for a nonlinear interface problem: A bounded body with a viscoplastic or plastic material behaviour is surrounded by an elastic body. The nonlinearity is treated by the finite element method while large parts of the linear elastic body are approximated using the boundary element method. Based on the a posteriori error estimates we derive an algorithm for the adaptive mesh refinement of the boundary elements and the finite elements. Its implementation is documented and numerical examples are included.     

An Adaptive Newton Algorithm for Optimal Control Problems with Application to Optimal Electrode Design

In this work, we present an adaptive Newton-type method to solve nonlinear constrained optimization problems, in which the constraint is a system of partial differential equations discretized by the finite element method. The adaptive strategy is based on a goal-oriented a posteriori error estimation for the discretization and for the iteration error. The iteration error stems from an inexact solution of the nonlinear system of first-order optimality conditions by the Newton-type method. This strategy allows one to balance the two errors and to derive effective stopping criteria for the Newton iterations. The algorithm proceeds with the search of the optimal point on coarse grids, which are refined only if the discretization error becomes dominant. Using computable error indicators, the mesh is refined locally leading to a highly efficient solution process. The performance of the algorithm is shown with several examples and in particular with an application in the neurosciences: the optimal electrode design for the study of neuronal networks.     

中子输运方程误差估计及自适应计算

 本文研究了全离散方法求解二维中子输运方程的有限元自适应算法, 角度变量用离散纵坐标方法展开, 空间变量用间断元方法求解. 基于间断元方法给出了空间离散的残量型后验误差估计. 在后验误差估计的基础上, 我们设计了自适应有限元算法.由残量型后验估计可以给出局部加密网格的自适应算法. 最后, 我们给出了数值算例来验证我们的理论结果.     

A posteriori error analysis of the discontinuous finite element methods for first order hyperbolic problems

In this paper, we consider the a posteriori error analysis of discontinuous Galerkin finite element methods for the steady and nonsteady first order hyperbolic problems with inflow boundary conditions. We establish several residual-based a posteriori error estimators which provide global upper bounds and a local lower bound on the error. Further, for nonsteady problem, we construct a fully discrete discontinuous finite element scheme and derive the a posteriori error estimators which yield global upper bound on the error in time and space. Our a posteriori error analysis is based on the mesh-dependent a priori estimates for the first order hyperbolic problems. These a posteriori error analysis results can be applied to develop the adaptive discontinuous finite element methods.     

Numerical analysis of the primal problem of elastoplasticity with hardening

Summary. The finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time-step in an elastoplastic evolution problem. In this paper, we analyse the finite element discretization and prove a priori and a posteriori error estimates for variational inequalities corresponding to the primal formulation of (Hencky) plasticity. The finite element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear resp. piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which shows linear convergence as the mesh-size tends to zero, provided the exact displacement field u is smooth. Near the boundary of the plastic domain, which is unknown a priori, it is most likely that u is non-smooth. In this situation, automatic mesh-refinement strategies are believed to improve the quality of the finite element approximation. We suggest such an adaptive algorithm on the basis of a computable a posteriori error estimate. This estimate is reliable and efficient in the sense that the quotient of the error by the estimate and its inverse are bounded from above. The constants depend on the hardening involved and become larger for decreasing hardening. Received May 7, 1997 / Revised version received August 31, 1998     

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.     

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.     

An adaptive C~0IPG method for the Helmholtz transmission eigenvalue problem

The interior penalty methods using C~0 Lagrange elements(C~0 IPG) developed in the recent decade for the fourth order problems are an interesting topic at present. In this paper, we discuss the adaptive proporty of C~0 IPG method for the Helmholtz transmission eigenvalue problem. We give the a posteriori error indicators for primal and dual eigenfunctions, and prove their reliability and efficiency. We also give the a posteriori error indicator for eigenvalues and design a C~0 IPG adaptive algorithm. Numerical experiments show that this algorithm is efficient and can get the optimal convergence rate.     

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.     

A Posteriori Error Estimates for Axisymmetric and Nonlinear Problems

We propose and examine the primal and dual finite element method for solving an axially symmetric elliptic problem with mixed boundary conditions. We derive an a posteriori error estimate and generalize the method used for a nonlinear elliptic problem. Finally, an a posteriori error estimate for a nonlinear parabolic problem based on the concept of hierarchical finite element basis functions is introduced.     

Quasi‐optimality of the adaptive finite element method for the eigenvalue problem arising from the vibration frequencies of the cavity flow

In this article, we present a posteriori error analysis for the regularization formulation of the eigenvalue problem arising from the vibration frequencies of the cavity flow. The quasi‐optimality of the adaptive finite element method is also proved for the single eigenvalues under the Dörfler's marking strategy without marking the oscillation terms and enforcing the so‐called interior node property. Numerical examples illustrate the quasi‐optimality of the adaptive finite element method. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 900–922, 2015