Dirichlet control problems with energy regularization governed by non-coercive elliptic equations

The present study investigates a linear-quadratic Dirichlet control problem governed by a non-coercive elliptic equation posed on a possibly non-convex polygonal domain. Tikhonov regularization is carried out in an energy seminorm. The regularity of the solutions is established in appropriate weighted Sobolev spaces, and the finite element discretization of the problem is analyzed. In order to recover the optimal rate of convergence in polygonal non-convex domains, graded meshes are required. In addressing this particular problem, it is also necessary to introduce a discrete projection in the sense of $H^{1/2}(Γ)$ to deal with the non-homogeneous boundary condition. A thorough examination of the approximation properties of the discrete controls reveals that the discrete problems are strongly convex uniformly with respect to the discretization parameter. All these ingredients lead to optimal error estimates. Practical computational considerations and numerical examples are discussed at the end of the paper.