Adaptive Finite Element Approximation for a Constrained Optimal Control Problem via Multi-meshes

In this paper, we study adaptive finite element approximation schemes for a constrained optimal control problem. We derive the 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.     

A posteriori error estimates for discontinuous Galerkin approximation of non-stationary convection-diffusion optimal control problems

In this paper, we investigate a discontinuous Galerkin finite element approximation of non-stationary convection dominated diffusion optimal control problems with control constraints. The state variable is approximated by piecewise linear polynomial space and the control variable is discretized by variational discretization concept. Backward Euler method is used for time discretization. With the help of elliptic reconstruction technique residual type a posteriori error estimates are derived for state variable and adjoint state variable, which can be used to guide the mesh refinement in the adaptive algorithm. Numerical experiment is presented, which indicates the good behaviour of the a posteriori error estimators.     

A Posteriori Error Estimates of Recovery Type for Distributed Convex Optimal Control Problems

In this paper, we derive a posteriori error estimates of recovery type, and present the superconvergence analysis for the finite element approximation of distributed convex optimal control problems. We provide a posteriori error estimates of recovery type for both the control and the state approximation, which are generally equivalent. Under some stronger assumptions, they are further shown to be asymptotically exact. Such estimates, which are apparently not available in the literature, can be used to construct adaptive finite element approximation schemes and as a reliability bound for the control problems. Numerical results demonstrating our theoretical results are also presented in this paper.     

Some a Posteriori Error Estimators for p-Laplacian Based on Residual Estimation or Gradient Recovery

In this paper, we first derive a posteriori error estimators of residual type for the finite element approximation of the p-Laplacian, and show that they are reliable, and efficient up to higher order terms. We then construct some a posteriori error estimators based on gradient recovery. We further compare the two types of a posteriori error estimators. It is found that there exist some relationships between the two types of estimators, which are similar to those held in the case of the Laplacian. It is shown that the a posteriori error estimators based on gradient recovery are equivalent to the discretization error in a quasi-norm provided the solution is sufficiently smooth and mesh is uniform. Under stronger conditions, superconvergnece properties have been established for the used gradient recovery operator, and then some of the gradient recovery based estimates are further shown to be asymptotically exact to the discretization error in a quasi-norm. Numerical results demonstrating these a posteriori estimates are also presented.     

An Adaptive Finite Element Method for the Wave Scattering with Transparent Boundary Condition

Consider the acoustic wave scattering by an impenetrable obstacle in two dimensions. The model is formulated as a boundary value problem for the Helmholtz equation with a transparent boundary condition. Based on a duality argument technique, an a posteriori error estimate is derived for the finite element method with the truncated Dirichlet-to-Neumann boundary operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of boundary operator which decays exponentially with respect to the truncation parameter. A new adaptive finite element algorithm is proposed for solving the acoustic obstacle scattering problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are marked through the finite element discretization error. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method.     

Application of a posteriori error estimation to finite element simulation of compressible Navier-Stokes flow

In this paper, we discuss how to improve the adaptive finite element simulation of compressible Navier-Stokes flow via a posteriori error estimate analysis. We use the moving space-time finite element method to globally discretize the time-dependent Navier-Stokes equations on a series of adapted meshes. The generalized compressible Stokes problem, which is the Stokes problem in its most generalized form, is presented and discussed. On the basis of the a posteriori error estimator for the generalized compressible Stokes problem, a numerical framework of a posteriori error estimation is established corresponding to the case of compressible Navier-Stokes equations. Guided by the a posteriori errors estimation, a combination of different mesh adaptive schemes involving simultaneous refinement/unrefinement and point-moving are applied to control the finite element mesh quality. Finally, a series of numerical experiments will be performed involving the compressible Stokes and Navier-Stokes flows around different aerodynamic shapes to prove the validity of our mesh adaptive algorithms.     

A posteriori error control in radiative transfer

This note introduces a finite element approach for a radiative transfer model equation which allows for a posteriori error control and thus opens up the way for adaptive grid refinement strategies. Apart from a posteriori error estimates for the mean intensity, we demonstrate corresponding results for the intensity and compare our error estimators with a new, simple and rather promising error indicator for hyperbolic problems. We specifically emphasize that our ‘pure’ finite element technique is equivalent to the well-established discrete ordinates method favored by many users.     

A posteriori error estimation in sensitivity analysis

We present a posteriori error estimators suitable for automatic mesh refinement in the numerical evaluation of sensitivity by means of the finite element method. Both diffusion (Poisson-type) and elasticity problems are considered, and the equivalence between the true error and the proposed error estimator is proved. Application to shape sensitivity is briefly addressed.     

发展型对流占优扩散方程的FD-SD法的后验误差估计及空间网格调节技术

０．引言 流线扩散法（ｓｔｒｅａｍｌｉｎｅ　ｄｉｆｆｕｓｉｏｎ ｍｅｔｈｏｄ,简称 ＳＤ法）是由Ｈｕｇｈｅｓ和 Ｂｒｏｏｋｓ在１９８０年前后提出的一种数值求解对流占优扩散问题的新型有限元算法．随后,Ｊｏｈｎｓｏｎ和 Ｎａｖｅｒｔ把ＳＤ法推广到发展型对流扩散问题．这一方法因其兼具良好的数值稳定性和高阶精度,近年来在理论与实践方面都得到了很大发展． 对于发展型对流扩散问题的ＳＤ法均采用时空有限元,即把时间、空间同等对待,这样做虽然使关于时间、空间的精度很好地统一起来,但与传统的有限元相比,由于维数增加,计…     

Adaptive finite element heterogeneous multiscale method for homogenization problems

In this paper we present an a posteriori error analysis for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. Unlike standard finite element methods, our discretization scheme relies on macro- and microfinite elements. The desired macroscopic solution is obtained by a suitable averaging procedure based on microscopic data. As the macroscopic data (such as the macroscopic diffusion tensor) are not available beforehand, appropriate error indicators have to be defined for designing adaptive methods. We show that such indicators based only on the available macro- and microsolutions (used to compute the actual macrosolution) can be defined, allowing for a macroscopic mesh refinement strategy which is both reliable and efficient. The corresponding a posteriori estimates for the upper and lower bound are derived in the energy norm. In the case of a uniformly oscillating tensor, we recover the standard residual-based a posteriori error estimate for the finite element method applied to the homogenized problem. Numerical experiments confirm the efficiency and reliability of the adaptive multiscale method.     

Adaptive mixed finite element method for Reissner–Mindlin plate

Mixed finite element methods are designed to overcome shear locking phenomena observed in the numerical treatment of Reissner–Mindlin plate models. Automatic adaptive mesh-refining algorithms are an important tool to improve the approximation behavior of the finite element discretization. In this paper, a reliable and robust residual-based a posteriori error estimate is derived, which evaluates a t-depending residual norm based on results in [D. Arnold, R. Falk, R. Winther, Math. Modell. Numer. Anal. 31 (1997) 517–557]. The localized error indicators suggest an adaptive algorithm for automatic mesh refinement. Numerical examples prove that the new scheme is efficient.     

Application of a posteriori error estimation to finite element simulation of incompressible Navier-Stokes flow

The main goal of this paper is to study adaptive mesh techniques, using a posteriori error estimates, for the finite element solution of the Navier-Stokes equations modeling steady and unsteady flows of an incompressible viscous fluid. Among existing operator splitting techniques, the θ-scheme is used for time integration of the Navier-Stokes equations. Then, a posteriori error estimates, based on the solution of a local system for each triangular element, are presented in the framework of the generalized incompressible Stokes problem, followed by its practical application to the case of incompressible Navier-Stokes problem. Hierarchical mesh adaptive techniques are developed in response to the a posteriori error estimation. Numerical simulations of viscous flows associated with selected geometries are performed and discussed to demonstrate the accuracy and efficiency of our methodology.     

Adaptive modeling of turbulent flow with residual based turbulent kinetic energy dissipation

In this paper we first review our recent work on a new framework for adaptive turbulence simulation: we model turbulence by weak solutions to the Navier–Stokes equations that are wellposed with respect to mean value output in the form of functionals, and we use an adaptive finite element method to compute approximations with a posteriori error control based on the error in the functional output. We then derive a local energy estimate for a particular finite element method, which we connect to related work on dissipative weak Euler solutions with kinetic energy dissipation due to lack of local smoothness of the weak solutions. The ideas are illustrated by numerical results, where we observe a law of finite dissipation with respect to a decreasing mesh size.     

An adaptive finite element method for evolutionary convection dominated problems

We present an adaptive finite element method for evolutionary convection–diffusion problems. The algorithm is based on an a posteriori indicator of the size of the oscillations displayed by the finite element approximation. The procedure is able to refine or coarsen dynamically the mesh adjusting it automatically to evolving layers. The method produces nearly non-oscillatory approximations in the convection dominated regime. We check the performance of the adaptive method with some numerical experiments.     

Adaptive dimension splitting algorithm for three-dimensional elliptic equations

An adaptive dimension splitting algorithm for three-dimensional (3D) elliptic equations is presented in this paper. We propose residual and recovery-based error estimators with respect to X?Y plane direction and Z direction, respectively, and construct the corresponding adaptive algorithm. Two-sided bounds of the estimators guarantee the efficiency and reliability of such error estimators. Numerical examples verify their efficiency both in estimating the error and in refining the mesh adaptively. This algorithm can be compared with or even better than the 3D adaptive finite element method with tetrahedral elements in some cases. What is more, our new algorithm involves only two-dimensional mesh and one-dimensional mesh in the process of refining mesh adaptively, and it can be implemented in parallel.     

基于模糊控制的自适应有限元网格自动剖分

本文采用模糊控制策略，对有限元网格划分过程中难以控制的因素－网格密度作出自适应控制。同时借鉴先验自适应的思想，在后验控制过程中综合考虑了边界属性的影响，避免了较多层次的网格局部细化与退化，采用此方法，不仅保证了计算精度，而且可以提高运算效率。     

Adaptive error control for multigrid finite element

We consider the problem of adaptive error control in the finite element method including the error resulting from, inexact solution of the discrete equations. We prove a posteriori error estimates for a prototype elliptic model problem discretized by the finite element with a canomical multigrid algorithm. The proofs are based on a combination of so-called strong stability and, the orthogonality inherent in both the finite element method can the multigrid algorithm.     

A RT Mixed FEM/DG Scheme for Optimal Control Governed by Convection Diffusion Equations

In this paper, we provide a numerical scheme—RT mixed FEM/DG scheme for the constrained optimal control problem governed by convection dominated diffusion equations. A priori and a posteriori error estimates are obtained for both the state, the co-state and the control. The adaptive mesh refinement can be applied indicated by a posteriori error estimator provided in this paper. Numerical examples are presented to illustrate the theoretical analysis.     

Adaptive methods for frictionless contact problems

In this paper a posteriori error indicators for frictionless contact problems are presented. In detail, error indicators relying on superconvergence properties and error estimators based on duality principles are investigated. Applications are to 3D solids under the hypothesis of non-linear elastic material behaviour associated with finite deformations. A penalization technique is applied to enforce multilateral boundary conditions due to contact. The approximate solution of the problem is obtained by using the finite element method. Several numerical results are reported to show the applicability of the adaptive algorithm to the considered problems.     

A posteriori error estimates for nonconforming approximations of Steklov eigenvalue problems

This paper deals with a posteriori error estimators for the non conforming Crouzeix-Raviart finite element approximations of the Steklov eigenvalue problem. First, we define an error estimator of the residual type which can be computed locally from the approximate eigenpair and we prove the equivalence between this estimator and the broken energy norm of the error with constants independent of the corresponding eigenvalue. Next, we prove that edge residuals dominate the volumetric part of the residual and that the volumetric part of the residual terms dominate the normal component of the jumps of the discrete fluxes across interior edges. Finally, based on these results, we introduce two simpler equivalent error estimators. The analysis shows that these a posteriori error estimates are optimal up to higher order terms and that may be used for the design of adaptive algorithms.