Optimal Control of Transmission Problems and Numerical Approximation by Adaptive FEM-BEM Coupling

Abstract We present a method for the numerical approximation of an optimal control problem constrained by a full-space transmission problem. The problem can be controlled by the interior source term, the jump over the interface, the jump of the flux over the interface, or any combination thereof. We complete the first-order optimality condition by the Johnson–Nédélec coupling, and a Lagrange multiplier which corresponds to the Bielak–MacCamy coupling. Our final formulation fulfills the LBB conditions on the continuous as well as discrete level, without restrictions. We develop a reliable a posteriori error estimator, propose an adaptive algorithm, and show its rate optimality. Numerical experiments confirm our theoretical findings.