Convergence and quasi-optimality of <inline-formula><tex-math id="M1">$ L^2- $</tex-math></inline-formula>norms based an adaptive finite element method for nonlinear optimal control problems

Abstract

<p style='text-indent:20px;'>This paper aims at investigating the convergence and quasi- optimality of an adaptive finite element method for control constrained nonlinear elliptic optimal control problems. We derive a posteriori error estimation for both the control, the state and adjoint state variables under controlling by <inline-formula><tex-math id="M2">$ L^2 $</tex-math></inline-formula>-norms where bubble function is a wonderful tool to deal with the global lower error bound. Then a contraction is proved before the convergence is proposed. Furthermore, we find that if keeping the grids sufficiently mildly graded, we can prove the optimal convergence and the quasi-optimality for the adaptive finite element method. In addition, some numerical results are presented to verify our theoretical analysis.</p>