Finite Element Methods of Dirichlet Boundary Optimal Control Problems With Weakly Imposed Boundary Conditions

In the present work, we consider Symmetric Interior Penalty Galerkin (SIPG) method to approximate the solution to Dirichlet optimal control problem governed by a linear advection-diffusion-reaction equation on a convex polygonal domain. The main feature of the method is that Dirichlet boundary conditions enter naturally into bilinear form and the finite element analysis can be performed in the standard setting. Another advantage of the method is that the method is stable and can be of arbitrary high degree. We show existence and uniqueness of the analytical and discrete solutions of the problem and derive optimal error estimates for the control on general convex polygonal domains. Finally, we support our main results and highlight some of the features of the method with the several numerical examples in one and two dimensions. We also investigate numerically the performance of the method for advection-dominated problems