LAUSR

In the presence of re-entrant corners, the solution of PDE-constrained optimization problems, in general, has singular components, even when the given data are smooth. Consequently, the use of standard finite element discretization gives rise to the so-called ’pollution effect’. This means that only a reduced convergence rate, compared to the best approximation error, is obtained. We discuss how optimal convergence rates in weighted norms can be achieved using the idea of energy corrected finite element methods, applied to optimal Dirichlet boundary control problem in the energy space. We present optimal error estimates in weighted norms for the state variable and for the control. Several numerical examples illustrate the obtained theoretical results.