A PRIORI ERROR ESTIMATE AND SUPERCONVERGENCE ANALYSIS FOR AN OPTIMAL CONTROL PROBLEM OF BILINEAR TYPE

In this paper, we investigate a priori error estimates and superconvergence properties for a model optimal control problem of bilinear type, which includes some parameter estimation application. The state and co-state are discretized by piecewise linear functions and control is approximated by piecewise constant functions. We derive a priori error estimates and superconvergence analysis for both the control and the state approximations. We also give the optimal L^2-norm error estimates and the almost optimal L^∞-norm estimates about the state and co-state. The results can be readily used for constructing a posteriori error estimators in adaptive finite element approximation of such optimal control problems.     

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.     

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.     

ESTIMATOR COMPETITION FOR POISSON PROBLEMS

We compare 13 different a posteriori error estimators for the Poisson problem with lowest-order finite element discretization. Residual-based error estimators compete with a wide range of averaging estimators and estimators based on local problems. Among our five benchmark problems we also look on two examples with discontinuous isotropic diffusion and their impact on the performance of the estimators. （Supported by DFG Research Center MATHEON.）     

A POSTERIORI ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS OF THE CAHN-HILLIARD EQUATION AND THE HELE-SHAW FLOW

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut ＋ △（ε△Au-ε^-1f（u）） = 0. It is shown that the a posteriori error bounds depends on ε^-1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct at2 adaptive algorithm for computing the solution of the Cahn- Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.     

VARIATIONAL DISCRETIZATION FOR OPTIMAL CONTROL GOVERNED BY CONVECTION DOMINATED DIFFUSION EQUATIONS

In this paper, we study variational discretization for the constrained optimal control problem governed by convection dominated diffusion equations, where the state equation is approximated by the edge stabilization Galerkin method. A priori error estimates are derived for the state, the adjoint state and the control. Moreover, residual type a posteriori error estimates in the L^2-norm are obtained. Finally, two numerical experiments are presented to illustrate the theoretical results.     

HANGING NODES IN THE UNIFYING THEORY OF A POSTERIORI FINITE ELEMENT ERROR CONTROL

A unified a posteriori error analysis has been developed in [18, 21-23] to analyze the finite element error a posteriori under a universal roof. This paper contributes to the finite element meshes with hanging nodes which are required for local mesh-refining. The twodimensional 1-irregular triangulations into triangles and parallelograms and their combinations are considered with conforming and nonconforming finite element methods named after or by Courant, Q1, Crouzeix-Raviart, Poisson, Stokes and Navier-Lamé equations Han, Rannacher-Turek, and others for the The paper provides a unified a priori and a posteriori error analysis for triangulations with hanging nodes of degree ≤ 1 which are fundamental for local mesh refinement in self-adaptive finite element discretisations.     

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.     

REDUCED BASIS METHOD FOR PARAMETRIZED ELLIPTIC ADVECTION-REACTION PROBLEMS

In this work we consider the Reduced Basis method for the solution of parametrized advection-reaction partial differential equations. For the generation of the basis we adopt a stabilized finite element method and we define the Reduced Basis method in the ＂primal- dual＂ formulation for this stabilized problem. We provide a priori Reduced Basis error estimates and we discuss the effects of the finite element approximation on the Reduced Basis error. We propose an adaptive algorithm, based on the a posteriori Reduced Basis error estimate, for the selection of the sample sets upon which the basis are built; the idea leading this algorithm is the minimization of the computational costs associated with the solution of the Reduced Basis problem. Numerical tests demonstrate the efficiency, in terms of computational costs, of the ＂primal-dual＂ Reduced Basis approach with respect to an ＂only primal＂ one. Parametrized advection-reaction partial differential equations, Reduced Basis method, ＂primal-dual＂ reduced basis approach, Stabilized finite element method, a posteriori error estimation.     

A POSTERIORI ENERGY-NORM ERROR ESTIMATES FOR ADVECTION-DIFFUSION EQUATIONS APPROXIMATED BY WEIGHTED INTERIOR PENALTY METHODS

We propose and analyze a posteriori energy-norm error estimates for weighted interior penalty discontinuous Galerkin approximations of advection-diffusion-reaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diffusion tensor and are used to formulate the consistency terms in the discontinuous Galerkin method. The error upper bounds, in which all the constants are specified, consist of three terms： a residual estimator which depends only on the elementwise fluctuation of the discrete solution residual, a diffusive flux estimator where the weights used in the method enter explicitly, and a non-conforming estimator which is nonzero because of the use of discontinuous finite element spaces. The three estimators can be bounded locally by the approximation error. A particular attention is given to the dependency on problem parameters of the constants in the local lower error bounds. For moderate advection, it is shown that full robustness with respect to diffusion heterogeneities is achieved owing to the specific design of the weights in the discontinuous Galerkin method, while diffusion anisotropies remain purely local and impact the constants through the square root of the condition number of the diffusion tensor. For dominant advection, it is shown, in the spirit of previous work by Verfiirth on continuous finite elements, that the local lower error bounds can be written with constants involving a cut-off for the ratio of local mesh size to the reciprocal of the square root of the lowest local eignevalue of the diffusion tensor.     

ON RESIDUAL-BASED A POSTERIORI ERROR ESTIMATORS FOR LOWEST-ORDER RAVIART-THOMAS ELEMENT APPROXIMATION TO CONVECTION-DIFFUSION-REACTION EQUATIONS

A new technique of residual-type a posteriori error analysis is developed for the lowest- order Raviart-Thomas mixed finite element discretizations of convection-diffusion-reaction equations in two- or three-dimension. Both centered mixed scheme and upwind-weighted mixed scheme are considered. The a posteriori error estimators, derived for the stress variable error plus scalar displacement error in L_2-norm, can be directly computed with the solutions of the mixed schemes without any additional cost, and are proven to be reliable. Local efficiency dependent on local variations in coefficients is obtained without any saturation assumption, and holds from the cases where convection or reaction is not present to convection- or reaction-dominated problems. The main tools of the analysis are the postprocessed approximation of scalar displacement, abstract error estimates, and the property of modified Oswald interpolation. Numerical experiments are carried out to support our theoretical results and to show the competitive behavior of the proposed posteriori error estimates.     

A POSTERIORI ESTIMATOR OF NONCONFORMING FINITE ELEMENT METHOD FOR FOURTH ORDER ELLIPTIC PERTURBATION PROBLEMS

In this paper, we consider the nonconforming finite element approximations of fourth order elliptic perturbation problems in two dimensions. We present an a posteriori error estimator under certain conditions, and give an h-version adaptive algorithm based on the error estimation. The local behavior of the estimator is analyzed as well. This estimator works for several nonconforming methods, such as the modified Morley method and the modified Zienkiewicz method, and under some assumptions, it is an optimal one. Numerical examples are reported, with a linear stationary Cahn-HiUiard-type equation as a model problem.     

An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation

We derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary. The numerical solution is assumed to be defined on a Finite Element mesh, whose boundary vertices are located on the boundary of the continuous problem. No assumption is made on a geometrically fitting shape.

A posteriori error estimates are given in the energy norm and the -norm, and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation. Furthermore, a strategy is presented to compute the effect of the non-discretized part of the domain on the error starting from a coarse mesh. This especially implies that parts of the domain, where the measured error is small, stay non-discretized. The presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary, the reliable computation of the a posteriori estimates and a mesh adaptation based on Delaunay techniques. Numerical examples illustrate that errors outside the initial discretization will be detected.

     

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.     

Error analysis of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation

This article discusses a priori and a posteriori error estimates of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation. We use variational discretization concept to discretize the control variable and discontinuous piecewise linear finite elements to approximate the state and costate variable. Based on the error estimates of discontinuous Galerkin finite element method for the transport equation, we get a priori and a posteriori error estimates for the transport equation optimal control problem. Finally, two numerical experiments are carried out to confirm the theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1493–1512, 2017     

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.     

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.     

Discontinuous Galerkin finite element method for a nonlinear boundary value problem

In this paper, we investigate the a priori and a posteriori error estimates for the discontinuous Galerkin finite element approximation to a regularization version of the variational inequality of the second kind. We show the optimal error estimates in the DG-norm （stronger than the H1 norm） and the L2 norm, respectively. Furthermore, some residual-based a posteriori error estimators are established which provide global upper bounds and local lower bounds on the discretization error. These a posteriori analysis results can be applied to develop the adaptive DG methods.     

定常的Navier-Stokes方程的非线性Galerkin混合元法及其后验估计

提出了定常的Navier-Stokes方程的一种非线性Galerkin混合元法,并导出非线性Galerkin混合元解的存在性和误差估计及其后验误差估计.     

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.