Mesh refinement for shape optimization

Abstract. In this paper, sharp a posteriori error estimators are derived for a class of distributed elliptic optimal control problems. These error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems and are implemented in the adaptive approach. Our numerical results indicate that the sharp error estimators work satisfactorily in guiding the mesh adjustments and can save substantial computational work.Key words. mesh adaptivity, optimal control, a posteriori error estimate, finite element method AMS subject classifications. 49J20, 65N30PII. S0363012901389342 1. Introduction. Finite element approximation of optimal control problems has long been an important topic in engineering design work and has been extensively studied in the literature. There have been extensive theoretical and numerical studies for finite element approximation of various optimal control problems; see [2,12,13,15,20,23,37,44]. For instance, for the optimal control problems governed by some linear elliptic or parabolic state equations, a priori error estimates of the finite element approximation were established long ago; see, for example, [12,13,15,20,23,37]. Furthermore, a priori error estimates were established for the finite element approximation of some important flow control problems in [17] and [11]. A priori error estimates have also been obtained for a class of state constrained control problems in [43], though the state equation is assumed to be linear. In [29], this assumption has been removed by reformulating the control problem as an abstract optimization problem in some Banach spaces and then applying nonsmooth analysis. In fact, the state equation there can be a variational inequality.In recent years, the adaptive finite element method has been extensively investigated. Adaptive finite element approximation is among the most important means to boost the accuracy and efficiency of finite element discretizations. It ensures a higher density of nodes in a certain area of the given domain, where the solution is more difficult to approximate. At the heart of any adaptive finite element method is an a posteriori error estimator or indicator. The literature in this area is extensive. Some of the techniques directly relevant to our work can be found in [1,5,6,7,28,32,34,42,47]. It is our belief that adaptive finite element enhancement is one of the future directions to pursue in developing sophisticated numerical methods for optimal design problems.

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“…For η 4 , let w τ be set as in Lemma 3.5 with B τ =Ḡ| τ . It follows from (3.14), (2.1), (3.23), and (3.22) that…”

Section: Proof From the Optimality Conditions (23) We Deduce That mentioning

confidence: 99%

“…Initial attempts in this aspect have been reported only recently for some design problems; see, e.g., [3,4,38,41]. However, a posteriori error indicators of a heuristic nature are widely used in most applications.…”

Section: Introductionmentioning

confidence: 99%

Adaptive Finite Element Approximation for Distributed Elliptic Optimal Control Problems

Liu

et al. 2002

SIAM J. Control Optim.

243

126

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“…The literature in this area is huge. Some of techniques directly relevant to our work can be found in [2][3][4]8,10,14,[26][27][28].…”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimates for mixed finite element solutions of convex optimal control problems

Chen

Liu

2008

Journal of Computational and Applied Mathematics

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“…Finite element method is one of the efficient numerical methods for solving (1.1); the literature in this aspect is huge (see, e.g., [1][2][3]13]). Systematic introduction to the finite element method for partial differential equations and optimal control problems are available in, for example, [10,26,32].…”

Section: Introductionmentioning

confidence: 99%

Richardson Extrapolation and Defect Correction of Finite Element Methods for Optimal Control Problem

Liu¹

2010

JCM

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Asymptotic error expansions in H^-novva for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectangular meshes, the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied. The higher order numerical approximations are used to generate a posteriori error estimators for the finite element approximation.Mathematics subject classification: 65R20, 65M12, 65M60, 65N30, 76S05, 49J20.

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Mesh refinement for shape optimization

Cited by 28 publications

References 11 publications

Adaptive Finite Element Approximation for Distributed Elliptic Optimal Control Problems

Adaptive Finite Element Approximation for Distributed Elliptic Optimal Control Problems

A posteriori error estimates for mixed finite element solutions of convex optimal control problems

Richardson Extrapolation and Defect Correction of Finite Element Methods for Optimal Control Problem

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