turkmath.org

We investigate a numerical treatment of linear-quadratic boundary optimal control problems under bilateral bound constraints, which act on a part of boundary conditions, i.e. Dirichlet or Neumann boundary conditions. Boundary control problems plays an important role in many practical applications such as active boundary control of flows. For example, if one is interested in control by blowing and suction on parts of the boundary only, boundary controls with low regularity should be admissible which even may develop jump discontinuities. To handle discontinuity on the boundary numerically, we choose discontinuous Galerkin (DG) methods as a discretization method. DG methods have become recently as an alternative to the finite difference, finite volume and continuous finite element methods due the better convergence behavior, local mass conservation, flexibility in approximating rough solutions on complicated meshes and mesh adaptation. This talk will firstly focus on Neumann boundary control problems. An efficient and reliable residual-type error estimator is introduced by invoking data oscillations. Adaptive mesh refinement indicated by a posteriori error estimates is applied. Then, in the second part of the talk, Dirichlet boundary control problems will be introduced. The consideration of Dirichlet boundary control problems is more difficult than Neumann control from both the theoretical and the numerical point of view, because the Dirichlet boundary conditions do not directly enter the variational setting