A POSTERIORI ERROR ESTIMATE OF OPTIMAL CONTROL PROBLEM OF PDE WITH INTEGRAL CONSTRAINT FOR STATE

In this paper, we study adaptive finite element discretization schemes for an optimal control problem governed by elliptic PDE with an integral constraint for the state. We derive the equivalent a posteriori error estimator for the finite element approximation, which particularly suits adaptive multi-meshes to capture different singularities of the control and the state. Numerical examples are presented to demonstrate the efficiency of a posteriori error estimator and to confirm the theoretical results.     

SUPERCONVERGENCE AND A POSTERIORI ERROR ESTIMATES FOR BOUNDARY CONTROL GOVERNED BY STOKES EQUATIONS

In this paper, the superconvergence results are derived for a class of boundary con-trol problems governed by Stokes equations. We derive superconvergence results for boththe control and the state approximation. Base on superconvergence results, we obtainasymptotically exact a posteriori error estimates.     

SUPERCONVERGENCE ANALYSIS OF FINITE ELEMENT METHODS FOR OPTIMAL CONTROL PROBLEMS OF THE STATIONARY BENARD TYPE

In this paper, we consider the finite element approximation of the distributed optimal control problems of the stationary Benard type under the pointwise control constraint. The states and the co-states are approximated by polynomial functions of lowest-order mixed finite element space or piecewise linear functions and the control is approximated by piecewise constant functions. We give the superconvergence analysis for the control; it is proved that the approximation has a second-order rate of convergence. We further give the superconvergence analysis for the states and the co-states. Then we derive error estimates in L^∞-norm and optimal error estimates in L^2-norm.     

UNIFORMLY A POSTERIORI ERROR ESTIMATE FOR THE FINITE ELEMENT METHOD TO A MODEL PARAMETER DEPENDENT PROBLEM

This paper proposes a reliable and efficient a posteriori error estimator for the finite element methods for the beam problem. It is proved that the error can be bounded by the computable error estimator from above and below up to multiplicative constants that do neither depend on the meshsize nor on the thickness of the beam.     

LOCAL A PRIORI AND A POSTERIORI ERROR ESTIMATE OF TQC9 ELEMENT FOR THE BIHARMONIC EQUATION

In this paper, local a priori, local a posteriori and global a posteriori error estimates are obtained for TQC9 element for the biharmonic equation. An adaptive algorithm is given based on the a posteriori error estimates.     

A POSTERIORI ERROR ESTIMATE FOR BOUNDARY CONTROL PROBLEMS GOVERNED BY THE PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we discuss the a posteriori error estimate of the finite element approximation for the boundary control problems governed by the parabolic partial differential equations. Three different a posteriori error estimators are provided for the parabolic boundary control problems with the observations of the distributed state, the boundary state and the final state. It is proven that these estimators are reliable bounds of the finite element approximation errors, which can be used as the indicators of the mesh refinement in adaptive finite element methods.     

体积约束的非局部扩散问题的后验误差分析

本文针对体积约束的非局部扩散问题构造了新的后验误差指示器,证明了后验误差指示器的可靠性以及有效性.数值算例验证了理论结果.     

RICHARDSON EXTRAPOLATION AND DEFECT CORRECTION OF FINITE ELEMENT METHODS FOR OPTIMAL CONTROL PROBLEMS

Asymptotic error expansions in H^1-norm for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectan- gular meshes, the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied. The higher order numerical approximations are used to generate a posteriori error estimators for the finite element approximation.     

简化摩擦接触问题的对称弱超内罚间断Galerkin方法的先验和后验误差估计

本文讨论了简化摩擦接触问题的一类对称弱超内罚间断Galerkin方法.首先,在能量范数意义下得到最优先验误差估计.进一步,我们推导了一类残量型后验误差估计子,并证明了它的可靠性和有效性.     

A PRIORI AND A POSTERIORI ERROR ESTIMATES OF A WEAKLY OVER-PENALIZED INTERIOR PENALTY METHOD FOR NON-SELF-ADJOINT AND INDEFINITE PROBLEMS

In this paper, we study a weakly over-penalized interior penalty method for non-self- adjoint and indefinite problems. An optimal a priori error estimate in the energy norm is derived. In addition, we introduce a residual-based a posteriori error estimator, which is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to validate our theoretical analysis.     

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。     

SUPERCONVERGENCE ANALYSIS FOR CUBIC TRIANGULAR ELEMENT OF THE FINITE ELEMENT SUPERCONVERGENCE ANALYSIS FOR CUBIC TRIANGULAR ELEMENT OF THE FINITE ELEMENT

1. IntroductionAlthough we bed proved the superconvergence of quadratic triangular elem6ntsbefore 1985, the superconvergence reseaxch of k(k 2 3)-degree triangulax elemeds Onlyhas a few advances, e.g.,.Lin, Yan and Zhou (see [15]) Prove that the three degreeHerlnite elements possess superconvergence and Walilbin (see [5-7]) obtains a roughresult by using a fine interinr estimation, that is, the placement fUnCtion or its gradestmay have weak superconvergence in the lOcal syUUnetric points,e…     

RECOVERY A POSTERIORI ERROR ESTIMATES FOR GENERAL CONVEX ELLIPTIC OPTIMAL CONTROL PROBLEMS SUBJECT TO POINTWISE CONTROL CONSTRAINTS

Superconvergence and recovery a posteriori error estimates of the finite element ap- proximation for general convex optimal control problems are investigated in this paper. We obtain the superconvergence properties of finite element solutions, and by using the superconvergence results we get recovery a posteriori error estimates which are asymptotically exact under some regularity conditions. Some numerical examples are provided to verify the theoretical results.     

ON THE ERROR ESTIMATE OF LINEAR FINITE ELEMENT APPROXIMATION TO THE ELASTIC CONTACT PROBLEM WITH CURVED CONTACT BOUNDARY

1. PreliminaryThe error estimate of linear finite element approximation to the elastic contact problem with curved contact boundary was considered in [2], in which the authors obtainedthe error bound of O(hl) with a much complex proof, requirement of three times continuously differentiable for contact boundary and extra regularities of triangulation (c.f. [2, Theorem 3.3, p.149]). In this paper, we obtained the error bound of O(hi) withonly requirement of two times continuously differentiable…     

改进的椭圆型问题一阶渐近展开误差估计

1引言考虑下述多尺度椭圆问题:■(1)其中椭圆算子A_ε定义为A_ε=-■/(■x_i)(a_(ij)~ε■/(■x_j).(2)本文使用爱因斯坦求和约定,重复指标表示求和.系数a_(ij)~ε(x)=a_(ij)(x/ε)满足下列条件:     

AN IMPROVED ERROR ANALYSIS FOR FINITE ELEMENT APPROXIMATION OF BIOLUMINESCENCE TOMOGRAPHY

This paper is concerned with an ill-posed problem which results from the area of molecular imaging and is known as BLT problem. Using Tikhonov regularization technique, a quadratic optimization problem can be formulated. We provide an improved error estimate for the finite element approximation of the regularized optimization problem. Some numerical examples are presented to demonstrate our theoretical results.     

ADAPTIVE FINITE ELEMENT APPROXIMATION FOR A CLASS OF PARAMETER ESTIMATION PROBLEMS

In this paper, we study adaptive finite element discretisation schemes for a class of parameter estimation problem. We propose to efficient algorithms for the estimation problem use adaptive multi-meshes in developing We derive equivalent a posteriori error estimators for both the state and the control approximation, which particularly suit an adaptive multi-mesh finite element scheme. The error estimators are then implemented and tested with promising numerical results.     

椭圆最优控制问题分裂正定混合有限元方法的超收敛性分析

首先利用变分原理和最优化理论得到了原问题的等价最优性条件;其次构造了椭圆最优控制问题分裂正定混合有限元方法的逼近格式;再次通过引入一些重要的中间变量和投影算子,并利用投影算子的相关性质,结合分裂正定混合有限元本身的逼近结果,得到了椭圆最优控制问题分裂正定混合有限元方法的超收敛性;最后数值实验结果验证了所得理论结果的正确性.     

板问题三次样条有限元的超收敛及高精度组合公式

1．引言与问题提出双三次样条有限元是应用比较广泛的矩形板元之一，与其它协调元相比恻双三次Her－mite矩形元），除达到饱和精度O（h‘）外，计算量却小得多［‘］，因而受到人们的重视．目前关于这方面的研究工作已有许多［1，并但关于板弯曲问题的双三次样条有限元的超收敛结果，却尚未见到．本文应用双三次样条插值的逐项渐近展式，通过对有限元的双线性形式进行展开，得到了双三次样条有限元一、二阶偏导数的渐近展式及超收敛结果．考虑如下矩形板弯曲模型问题上述方程对应的双线性形式为对VV，VE”，满足其中，M，。为正常数，11…