Finite Element Analysis of Dirichlet Boundary Control Problems Governed by Certain PDEs.

In the first part, we study {\em a priori} error analysis of an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Poisson equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and co-state variables. We propose a finite element-based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. We present the analysis for \(L^2\) cost functional, but this analysis can also be extended to the gradient cost functional problem. The error estimates of optimal order in the energy norm are derived.

In the second part, we discuss {\em a posteriori} error analysis of the Dirichlet boundary optimal control problem governed by the Stokes equation. We develop a finite element discretization by using \(\mathbf{P}_1\) elements(in the fine mesh) for the velocity and control variable and \(P_0\) elements (in the coarse mesh) for the pressure variable. We present a new {\em a posteriori} error estimator for the control error.  We sketch out the proof of the estimator's reliability and efficiency.

In both parts of the talk, we verify the theoretical results by some numerical experiments.