Convergence and Quasi-Optimality of an Adaptive Finite Element Method for Optimal Control Problems on $$L^{2}$$ L 2 Errors

In this paper, we prove the convergence of an adaptive finite element method for optimal control problems on $$L^{2}$$L2 errors by keeping the meshes sufficiently mildly. In order to keep the meshes sufficiently mildly we need increasing the number of elements that are refined, moreover, we find that it will not compromise the quasi-optimality of the AFEM. In other words, we prove the quasi-optimality of the adaptive finite element algorithm in the present paper. In the end, we conclude this paper with some conclusions and future works.