Abstract
<abstract> This paper investigates the adaptive finite element method for an optimal control problem governed by a bilinear elliptic equation. We establish the finite element discrete scheme for the bilinear optimal control problem and use a dual argument, linearization method, bubble function, and new bubble function to obtain a posteriori error estimates. To prove the convergence and the quasi-optimality for adaptive finite element methods, we introduce the adaptive finite element algorithm, local perturbation, error reduction, discrete local upper bound, Dörfler property, dual argument method, and quasi orthogonality. A few numerical examples are given at the end of the paper to demonstrate our theoretical analysis. </abstract>
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