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In this work, an adaptive edge element method is developed for anH(curl)-elliptic constrained optimal control problem. We use the lowest-orderNedelec's edge elements of first family and the piecewise (element-wise)constant functions to approximate the state and the control, respectively, andpropose a new adaptive algorithm with error estimators involving bothresidual-type error estimators and lower-order data oscillations. By using alocal regular decomposition for H(curl)-functions and the standard bubblefunction techniques, we derive the a posteriori error estimates for theproposed error estimators. Then we exploit the convergence properties of theorthogonal $L^2$-projections and the mesh-size functions to demonstrate thatthe sequences of the discrete states and controls generated by the adaptivealgorithm converge strongly to the exact solutions of the state and control inthe energy-norm and $L^2$-norm, respectively, by first achieving the strongconvergence towards the solution to a limiting control problem.Three-dimensional numerical experiments are also presented to confirm ourtheoretical results and the quasi-optimality of the adaptive edge elementmethod.

An adaptive edge element method and its convergence for an electromagnetic constrained optimal control problem