非协调有限元逼近的梯度恢复型后验误差估计（英文）

本文给出二阶椭圆型方程的非协调有限元的梯度恢复型后验误差估计.后验误差估计是在Crouzeix-Raviart非协调有限单元上得到的,并且给出误差的上下界,更进一步可以证明所得的后验误差估计在拟一致网格上是渐近精确的,所以误差估计是可行的、有效的.上界证明过程依赖于Helmholtz分解,下界证明主要依赖bubble函数.数值结果验证了理论的正确性.     

非协调有限元误差的内部估计

§1.引言 Nitsche与Schatz等曾对经典的协调有限元近似解的误差作出了内部估计.Bra-mble,Schatz和Thomee在此基础上提出了用局部平均法得到超收敛的结果.近来     

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.     

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。     

达尔文模型的有限元后验误差估计

在三维有界单连通区域里,当没有高频现象或电流变化不快时,达尔文模型是麦克斯韦方程组的-个很好的逼近模型.本文考虑达尔文模型的自适应算法,这种方法以有限元后验误差分析为理论基础.本文提供了基于后验误差估计子的上界估计.     

一维有限元的h-p方法的后验误差估计

有限元的h-p方法,是指在增加有限元空间的维数时,既加密某些单元的网格,同时也增加某些单元的次数.对h-p方法,人们希望得到O(h~mp~(-n))(m,n>0)形状的误差估计.这种误差估计的结果包括了对传统的h方法以及p方法的结果.关于h-p方法的     

自适应有限元和后验误差估计——等价估计

近年来,自适应有限元计算得到了广泛的研究.后验误差估计是实现自适应计算的基础.70年代末,I. Babuska和 W.Rheinboldt给出了建立等价后验估计的一般原则和方法,在一维情形给出了完整的结果.80年代初,I.Babuska和A.Miller对平面弹性问题正方形双线性元,给出了一种后验估计的方法,并证明了其等价性和渐近准确     

一个非协调板元的误差估计

关于非协调板元的L_2—估计,文[1]曾进行了系统的研究,但对于Morely元和二个Fraeijs de Veubeke元(以下简称F.V.1元和F.V.2元)所得的结果并不是最优的。文[2]对Morely元又作了进一步的讨论,得到了最优的L_2估计及渐近最优的L_∞估计。本文将研究F.V.2元,得到了与[2]相仿的最优L_2-估计。但是,关于L_∞-估计,由于插值多     

非协调元的各向异性插值误差估计

The main aim of this paper is to study the local anisotropic interpolation error estimates. We show that the interpolation of a nonconforming element satisfy the anisotropic property for both the second and fourth order problems.     

A Posteriori Error Estimates on Irregular Grids Based on Gradient Recovery

In this article, an equilibrated gradient recovery error estimator is introduced and analyzed. Regional and global error bounds are established under the equilibrium condition. Furthermore, the error estimator based on the ZZ patch recovery technique is analyzed theoretically. Stability and consistent properties are proved under mild assumptions. All results are valid for arbitrary grids.     

A Posteriori Error Estimators of Gradient Recovery Type for Fem of a Model Optimal Control Problem

In this paper, we derive an a posteriori error estimator of gradient recovery type for a model optimal control problem. We show that the a posteriori error estimator is equivalent to the discretization error in a norm of energy type on general meshes. Furthermore, when the solution of the control problem is smooth and the meshes are uniform, it is shown to be asymptotically exact.     

双线形元的各向异性后验误差估计

The main aim of this paper is to give an anisotropic posteriori error estimator. We firstly study the convergence of bilinear finite element for the second order problem under anisotropic meshes.By using some novel approaches and techniques,the optimal error estimates and some superconvergence results are obtained without the regularity assumption and quasi-uniform assumption requirements on the meshes.Then,based on these results, we give an anisotropic posteriori error estimate for the second problem.     

线性对流占优扩散方程的后验误差估计

对于线性对流占优扩散方程,采用特征线有限元方法离散时间导数项和对流项,用分片线性有限元离散空间扩散项,并给出了一致的后验误差估计,其中估计常数不依赖与扩散项系数。     

Recovery Type a Posteriori Error Estimates of Fully Discrete Finite Element Methods for General Convex Parabolic Optimal Control Problems

This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints. The time discretization is based on the backward Euler method. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. We derive the superconvergence properties of finite element solutions. By using the superconvergence results, we obtain recovery type a posteriori error estimates. Some numerical examples are presented to verify the theoretical results.     

各向异性网格下具有数值积分的非协调有限元逼近

In this paper we mainly discuss the nonconforming finite element method for second order elliptic boundary value problems on anisotropic meshes.By changing the discretization form(i.e.,by use of numerical quadrature in the procedure of computing the left load),we obtain the optimal estimate O(h),which is as same as in the traditional finite element analysis when the load f∈H~1(Ω)∩C~0(Ω)which is weaker than the previous studies.The results obtained in this paper are also valid to the conforming triangular element and nonconforming Carey's element.     

A Posteriori Error Estimators of Gradient Recovery Type for Elliptic Obstacle Problems

In this paper, we present a posteriori error estimates of gradient recovery type for elliptic obstacle problems. The a posteriori error estimates provide both lower and upper error bounds. It is shown to be equivalent to the discretization error in an energy type norm for general meshes. Furthermore, when the solution is smooth and the mesh is uniform, it is shown to be asymptotically exact. Some numerical results which demonstrate the theoretical results are also reported in this paper.     

A Review of Unified a Posteriori Finite Element Error Control

This paper aims at a general guideline to obtain a posteriori error estimates for the finite element error control in computational partial differential equations. In the abstract setting of mixed formulations, a generalised formulation of the corresponding residuals is proposed which then allows for the unified estimation of the respective dual norms. Notably, this can be done with an approach which is applicable in the same way to conforming, nonconforming and mixed discretisations. Subsequently, the unified approach is applied to various model problems. In particular, we consider the Laplace, Stokes, Navier-Lamé, and the semi-discrete eddy current equations.     

一类非线性双曲型方程的非协调有限元方法

研究了一类非线性双曲型方程的非协调有限元方法,在不需要传统的Ritz投影的情况下,得到了半离散格式下的误差估计及超收敛结果.     

对流扩散方程双线性有限元解的一致估计

证明了对流扩散方程的半离散双线性Galerkin有限元方法关于小参数ε一致的最优阶误差估计,其中的常数只与初值和右端函数有关,而与小参数ε无关.