Adaptive Finite Element Approximation for Distributed Elliptic Optimal Control Problems

Abstract

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.

Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems

“…On the other hand, considerably less work has been done with regard to optimal control problems for PDEs. The so-called goal oriented dual weighted approach has been applied in the unconstrained case in [4,5] and to control constrained problems in [20,42], whereas residual-type a posteriori error estimators for control constrained problems have been derived and analyzed in [16,17,21,25,28,30,31]. Unlike the control constrained case, pointwise state constrained optimal control problems are much more difficult to handle due to the fact that the Lagrange multiplier for the state constraints lives in a measure space (see, e.g., [8,9,23,39]).…”

Goal Oriented Mesh Adaptivity for Mixed Control-State Constrained Elliptic Optimal Control Problems

“…Residual-type a posteriori error estimators for control constrained problems have been developed and analyzed in [13,14,18,20,23,26,27]. State constrained optimal control problems are more difficult to handle than control constrained ones, since the Lagrange multiplier for the state constraints typically lives in a measure space.…”

A Posteriori Estimates for Cost Functionals of Optimal Control Problems

“…Numerical solution of optimal control problems is usually based on applying specific iterative schemes to the system of optimality conditions, (e.g., the active set strategies or interior point methods; see, e.g., [6], [7] and references therein). Adaptive techniques for optimal control problems governed by PDEs are presented in, e.g., [4] and [9].…”