Boundary concentrated finite elements for optimal control problems with distributed observation

We consider the discretization of an optimal boundary control problem with distributed observation by the boundary concentrated finite element method. With an H(Ω) regular elliptic PDE on two-dimensional domains as constraint, we prove that the discretization error ‖u∗ − uh‖L2(Γ) decreases like N −δ, where N denotes the total number of unknowns. For the case δ = 1 in convex polygonal domains, the discretization error for h-FEM behaves like N−3/4, whereas for boundary concentrated FEM the discretization error behaves like N−1. This makes the boundary concentrated FEM favorable in comparison to h-FEM. This method is also suitable for treating piecewise defined data and a tracking functional acting only on a subdomain of Ω. We present several numerical results. Acknowledgment. This work was funded by Austrian Science Fund (FWF) grant P23484-N18.