Equivalent a Posteriori Error Estimates for a Constrained Optimal Control Problem Governed by Parabolic Equations

For the optimal control problems governed by linear elliptic or parabolic state equations, a priori error estimates of the finite element approximation were established long ago, see [1, 2, 3, 4, 5]. In order to obtain a numerical solution of acceptable accuracy for the optimal control problem, the finite element meshes have to be refined according to a mesh refinement scheme. Adaptive finite element approximation uses a posteriori indicators to guide the mesh refinement procedure. Only the area where the error indicator is larger will be refined so that a higher density of nodes is distributed over the area where the solution is difficult to be approximated. In this sense adaptive finite element approximation relies very much on the error indicator used. It has been recently found that suitable adaptive meshes can greatly reduce the control approximation errors, see [6, 7, 8, 9, 10]. If the computational meshes are not properly generated, then there may be large error around the singularities of the control, which cannot be removed later on. Furthermore in a constrained control problem, the optimal control and the state usually have very different regularities and their locations. This indicates that the all-in-one mesh strategy may be inefficient. Adaptive multi-meshes, that is, separate adaptive meshes which are adjusted according to different error indicators, are often necessary, see [11]. Using different adaptive meshes for the control and the state allows very coarse meshes to be used in solving the state and co-state equations. Thus much computational work can be saved because one of the major computational loads is to solve the state and co-state equations repeatedly. This can be clearly seen from numerical experiments in [11] and our numerical tests in Section 4.