Weak boundary penalization for Dirichlet boundary control problems governed by elliptic equations

This paper concerns the finite element approximation of Dirichlet boundary control problems governed by elliptic equations. Different from the existing literatures, in which standard finite element method, mixed finite element method or Robin penalization method are used to deal with the underlying problems, we adopt an alternative penalization approach introduced by Nitsche called weak boundary penalization. Compared with the above methods, our discrete scheme not only keeps consistency and avoids penalization error, but also can be analyzed and computed conveniently as Neumann boundary control problems. Based on the weak boundary penalization method, we establish a finite element approximation to the Dirichlet boundary control problems and derive the a priori error estimates for the control, state and adjoint state. Numerical experiments are provided to confirm our theoretical results.