Boundary concentrated finite elements for optimal boundary control problems of elliptic PDEs

We investigate the discretization of optimal boundary control problems for elliptic equations on two‐dimensional polygonal domains by the boundary concentrated finite element method. We prove that the discretization error $\parallel u^{*}‐u_h^{*}{\parallel }_{L^2(\Gamma )}$ decreases like N ^-1, where N is the total number of unknowns. This makes the proposed method favorable in comparison to the h‐version of the finite element method, where the discretization error behaves like N ^-3/4 for uniform meshes. Moreover, we present an algorithm that solves the discretized problem in almost optimal complexity. The paper is complemented with numerical results.