Optimal convergence of finite element approximation to an optimization problem with PDE constraint*Wei Gong was supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB 41000000), the National Key Basic Research Program (Grant No. 2018YFB0704304) and the National Natural Science Foundation of China (Grant No. 12071468, 11671391). Zhaojie Zhou was supported by the National Natural Science Foundation of China under Grant No. 11971276. (1st April 2022) - Gong, Wei; Tan, Zhiyu; Zhou, Zhaojie
Abstract: We study in this paper the optimal convergence of finite element approximation to an optimization problem with PDE constraint. Specifically, we consider an elliptic distributed optimal control problem without control constraints, which can also be viewed as a regularized inverse source problem. The main contributions are two-fold. First, we derive a priori and a posteriori error estimates for the optimization problems, under an appropriately chosen norm that allows us to establish an isomorphism between the solution space and its dual. These results yield error estimates with explicit dependence on the regularization parameter α so that the constants appeared in the derivation are independent of α . Second, we prove the contraction property and rate optimality for the adaptive algorithm with respect to the error estimator and solution errors between the adaptive finite element solutions and the continuous solutions. Extensive numerical experiments are presented that confirm our theoretical results.