New a posteriori L∞(L2) and L2(L2)-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems

We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in L^∞(J; L²Ω)-norm and L²(J; L²Ω)-norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important step towards developing a reliable adaptive mixed finite element approximation for optimal control problems. Finally, the performance of the posteriori error estimators is assessed by two numerical examples.     

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.     

Mixed Discontinuous Galerkin Time-Stepping Method for Linear Parabolic Optimal Control Problems

In this paper, we discuss the mixed discontinuous Galerkin (DG) finite element approximation to linear parabolic optimal control problems. For the state variables and the co-state variables, the discontinuous finite element method is used for the time discretization and the Raviart-Thomas mixed finite element method is used for the space discretization. We do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. We derive a priori error estimates for the lowest order mixed DG finite element approximation. Moveover, for the element of arbitrary order in space and time, we derive a posteriori $L^2(0, T ;L^2(Ω))$ error estimates for the scalar functions, assuming that only the underlying mesh is static. Finally, we present an example to confirm the theoretical result on a priori error estimates.     

A Posteriori Error Estimates for Finite Element Exterior Calculus: The de Rham Complex

Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk, and Winther includes a well-developed theory of finite element methods for Hodge–Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of a residual type for Arnold–Falk–Winther mixed finite element methods for Hodge–de Rham–Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwell’s equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by a unified treatment of the various Hodge–Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge–Laplacian.     

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.     

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.     

Recovery type superconvergence and a posteriori error estimates for control problems governed by Stokes equations

In this paper, we derive recovery type superconvergence analysis and a posteriori error estimates for the finite element approximation of the distributed optimal control governed by Stokes equations. We obtain superconvergence results and asymptotically exact a posteriori error estimates by applying two recovery methods, which are the patch recovery technique and the least-squares surface fitting method. Our results are based on some regularity assumption for the Stokes control problems and are applicable to the first order conforming finite element method with regular but nonuniform partitions.     

Superconvergence and a posteriori error estimates of RT1 mixed methods for elliptic control problems with an integral constraint

In this paper, we investigate the superconvergence property and a posteriori error estimates of mixed finite element methods for a linear elliptic control problem with an integral constraint. The state and co-state are approximated by the order k = 1 Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Approximations of the optimal control of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order h ². Moreover, we derive a posteriori error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.     

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.     

一阶双曲问题间断有限元的后验误差分析

一阶双曲问题的有限元后验误差估计至今没有得到很好的解决.本文对d维区域上一阶双曲问题的k次间断有限元逼近提出了一种新的后验误差分析方法, 进而建立了间断有限元解在DG范数下(强于L₂范数)基于误差余量型的后验误差估计. 数值计算验证了本文理论分析的有效性. 本文方法也适用于其他变分问题有限元逼近的后验误差分析.     

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.     

Elliptic reconstruction and a posteriori error estimates for parabolic optimal control problems

In this article, a semidiscrete finite element method for parabolic optimal control problems is investigate. By using elliptic reconstruction, a posteriori error estimates for finite element discretizations of optimal control problem governed by parabolic equations with integral constraints are derived.     

Superconvergence of mixed finite element semi-discretizations of two time-dependent problems

We will show that some of the superconvergence properties for the mixed finite element method for elliptic problems are preserved in the mixed semi-discretizations for a diffusion equation and for a Maxwell equation in two space dimensions. With the help of mixed elliptic projection we will present estimates global and pointwise in time. The results for the Maxwell equations form an extension of existing results. For both problems, our results imply that post-processing and a posteriori error estimation for the error in the space discretization can be performed in the same way as for the underlying elliptic problem.     

热传导对流问题的自适应最小二乘Galerkin/Petrov混合有限元法

对热传导对流问题提出了自适应Galerkin/Petrov最小二乘混合有限元法．该算法对任何速度和压力有限元空间的组合是相容和稳定的(不需要满足Babuka-Brezzi稳定性条件)．利用Verfürth的一般理论,得到了热传导对流问题的残量型的后验误差估计．最后通过几个数值算例验证了方法的有效性．     

Application of functional error estimates with mixed approximations to plane problems of linear elasticity

S.I. Repin and his colleagues’ studies addressing functional a posteriori error estimates for solutions of linear elasticity problems are further developed. Although the numerical results obtained for planar problems by A.V. Muzalevsky and Repin point to advantages of the adaptive approach used, the degree of overestimation of the absolute error increases noticeably with mesh refinement. This shortcoming is eliminated by using approximations typical of mixed finite element methods. A comparative analysis is conducted for the classical finite element approximations, mixed Raviart-Thomas approximations, and relatively recently proposed Arnold-Boffi-Falk mixed approximations. It is shown that the last approximations are the most efficient.     

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 posteriori error estimator competition for conforming obstacle problems

This article on the a posteriori error analysis of the obstacle problem with affine obstacles and Courant finite elements compares five classes of error estimates for accurate guaranteed error control. To treat interesting computational benchmarks, the first part extends the Braess methodology from 2005 of the resulting a posteriori error control to mixed inhomogeneous boundary conditions. The resulting guaranteed global upper bound involves some auxiliary partial differential equation and leads to four contributions with explicit constants. Their efficiency is examined affirmatively for five benchmark examples. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

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 finite element methods for nonlinear quadratic boundary optimal control problem

This paper is aimed at studying finite element discretization for a class of quadratic boundary optimal control problems governed by nonlinear elliptic equations. We derive a posteriori error estimates for the coupled state and control approximation. Such estimates can be used to construct a reliable adaptive finite element approximation for the boundary optimal control problem. Finally, we present a numerical example to confirm our theoretical results.     

Error estimates in elastoplasticity using a mixed method

In this paper, a mixed formulation and its discretization are introduced for elastoplasticity with linear kinematic hardening. The mixed formulation relies on the introduction of a Lagrange multiplier to resolve the non-differentiability of the plastic work function. The main focus is on the derivation of a priori and a posteriori error estimates based on general discretization spaces. The estimates are applied to several low-order finite elements. In particular, a posteriori estimates are expressed in terms of standard residual estimates. Numerical experiments are presented, confirming the applicability of the a posteriori estimates within an adaptive procedure.