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.     

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.     

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.     

双线性非协调有限元逼近的梯度恢复后验误差估计

给出了二阶椭圆方程的双线性非协调有限元逼近的梯度恢复后验误差估计.该误差估计是在Q_1非协调元上得到的,并给出了误差的上下界.进一步证明该误差估计在拟一致网格上是渐进精确地.证明依赖于clement插值和Helmholtz分解,数值结果验证了理论的正确性.     

A Posteriori Estimators for the FEM: Analysis of the Robustness of the Estimators for the Poisson Equation

This paper is a summary of our study on estimators for the error in the finite element solution given in [4,5,7,9,10]. We will persent the analysis of estimators for finite element solutions of the Poisson equation in the interior of meshes of triangles with straight edges, as well as the analysis at the boundary.     

Functional A Posteriori Error Estimates for Time-Periodic Parabolic Optimal Control Problems

This article is devoted to the a posteriori error analysis of multiharmonic finite element approximations to distributed optimal control problems with time-periodic state equations of parabolic type. We derive a posteriori estimates of the functional type, which are easily computable and provide guaranteed upper bounds for the state and co-state errors as well as for the cost functional. These theoretical results are confirmed by several numerical tests that show high efficiency of the a posteriori error bounds.     

A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation

In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.     

A Posteriori Error Estimates for Distributed Convex Optimal Control Problems

In this paper, we present an a posteriori error analysis for finite element approximation of distributed convex optimal control problems. We derive a posteriori error estimates for the coupled state and control approximations under some assumptions which hold in many applications. Such estimates, which are apparently not available in the literature, can be used to construct reliable adaptive finite element approximation schemes for control problems. Explicit estimates are obtained for some model problems which frequently appear in real-life applications.     

Gradient recovery type a posteriori error estimate for finite element approximation on non-uniform meshes

In this paper, we derive gradient recovery type a posteriori error estimate for the finite element approximation of elliptic equations. We show that a posteriori error estimate provide both upper and lower bounds for the discretization error on the non-uniform meshes. Moreover, it is proved that a posteriori error estimate is also asymptotically exact on the uniform meshes if the solution is smooth enough. The numerical results demonstrating the theoretical results are also presented in this paper.     

A Modified Polynomial Preserving Recovery and Its Applications to A Posteriori Error Estimates

<正>A modified polynomial preserving gradient recovery technique is proposed. Unlike the polynomial preserving gradient recovery technique,the gradient recovered with the modified polynomial preserving recovery(MPPR) is constructed element-wise, and it is discontinuous across the interior edges.One advantage of the MPPR technique is that the implementation is easier when adaptive meshes are involved.Superconvergence results of the gradient recovered with MPPR are proved for finite element methods for elliptic boundary problems and eigenvalue problems under adaptive meshes. The MPPR is applied to adaptive finite element methods to construct asymptotic exact a posteriori error estimates.Numerical tests are provided to examine the theoretical results and the effectiveness of the adaptive finite element algorithms.     

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.     

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.     

A Posteriori Error Estimates of Mixed Methods for Parabolic Optimal Control Problems

In this article, we shall give a brief review on the fully discrete mixed finite element method for general optimal control problems governed by parabolic equations. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. Furthermore, we derive a posteriori error estimates for the finite element approximation solutions of optimal control problems. Some numerical examples are given to demonstrate our theoretical results.     

Elliptic Reconstruction and a Posteriori Error Estimates for Fully Discrete Semilinear Parabolic Optimal Control Problems

This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems. Based on elliptic reconstruction approach introduced earlier by Makridakis and Nochetto [25], a residual based a posteriori error estimators for the state, co-state and control variables are derived. The space discretization of the state and co-state variables is done by using the piecewise linear and continuous finite elements, whereas the piecewise constant functions are employed for the control variable. The temporal discretization is based on the backward Euler method. We derive a posteriori error estimates for the state, co-state and control variables in the $L^\infty(0,T;L^2(\Omega))$-norm. Finally, a numerical experiment is performed to illustrate the performance of the derived estimators.     

A Posteriori Error Estimators and Adaptivity for Finite Element Approximation of the Non-Homogeneous Dirichlet Problem

Techniques are developed for a posteriori error analysis of the non-homogeneous Dirichlet problem for the Laplacian giving computable error bounds for the error measured in the energy norm. The techniques are based on the equilibrated residual method that has proved to be reliable and accurate for the treatment of problems with homogeneous Dirichlet data. It is shown how the equilibrated residual method must be modified to include the practically important case of non-homogeneous Dirichlet data. Explicit and implicit a posteriori error estimators are derived and shown to be efficient and reliable. Numerical examples are provided illustrating the theory.     

Residual Based a Posteriori Error Estimates for Convex Optimal Control Problems Governed by Stokes-Darcy Equations

In this paper, we derive a posteriori error estimates for finite element approximations of the optimal control problems governed by the Stokes-Darcy system. We obtain a posteriori error estimators for both the state and the control based on the residual of the finite element approximation. It is proved that the a posteriori error estimate provided in this paper is both reliable and efficient.     

美式期权定价中非局部问题有限元方法的整体超收敛与后验估计

我们利用积分恒等式与插值后处理技术 ,讨论美式期权非局部问题有限元方法关于 H 1-模的整体超收敛与后验估计 .     

A Priori Error Estimates for Obstacle Optimal Control Problem,Where the Obstacle Is the Control Itself

In this paper we deal with the convergence analysis of the finite element method for an elliptic penalized unilateral obstacle optimal control problem where the control and the obstacle coincide. Error estimates are established for both state and control variables. We apply a fixed point type iteration method to solve the discretized problem.To corroborate our error estimations and the efficiency of our algorithms, the convergence results and numerical experiments are illustrated by concrete examples.     

η%-Superconvergence of Finite Element Solutions and Error Estimators

This paper is a summary of our study on the superconvergence of the finite element solutions and error estimators. We will persent the analysis of %-superconvergence for finite element solutions of the Poisson equation in the interior of meshes of triangles with straight edges, as well as the analysis at the boundary. The %-superconvergence via local averaging will also be presented, and the error estimators are compared in the sense of %-superconvergence.