Crank-Nicolson finite difference schemes for parabolic optimal Dirichlet boundary control problem

In this paper, we adopt the optimize-then-discretize approach to solve parabolic optimal Dirichlet boundary control problem. First, we derive the first-order necessary optimality system, which includes the state, co-state equations and the optimality condition. Then, we propose Crank-Nicolson finite difference schemes to discretize the optimality system in 1D and 2D cases, respectively. In order to build the second order spatial approximation, we use the ghost points on the boundary in the schemes. We prove that the proposed schemes are unconditionally stable, compatible and second-order convergent in both time and space. To avoid solving the large coupled schemes directly, we use the iterative method. Finally, we present a numerical example to validate our theoretical analysis.

Residual based a posteriori error estimation for Dirichlet boundary control problems

We study a residual–based a posteriori error estimate for the solution of Dirichlet boundary control problem governed by a convection diffusion equation on a two dimensional convex polygonal domain, using the local discontinuous Galerkin (LDG) method with upwinding for the convection term. With the usage of LDG method, the control variable naturally exists in the variational form due to its mixed finite element structure. We also demonstrate the application of our a posteriori error estimator for the adaptive solution of these optimal control problems.

A Finite Element Method for Elliptic Dirichlet Boundary Control Problems

We consider the finite element discretization of an optimal Dirichlet boundary control problem for the Laplacian, where the control is considered in {H^{1/2}(\Gamma)}. To avoid computing the latter norm numerically, we realize it using the {H^{1}(\Omega)} norm of the harmonic extension of the control. We propose a mixed finite element discretization, where the harmonicity of the solution is included by a Lagrangian multiplier. In the case of convex polygonal domains, optimal error estimates in the {H^{1}} and {L_{2}} norm are proven. We also consider and analyze the case of control constrained problems.

Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem

We study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Laplace 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 costate variables. We propose a finite element based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. The analysis is presented in a combination for both the gradient and the L2 cost functional. A priori error estimates of optimal order in the energy norm is derived up to the regularity of the solution for both the cases. Theoretical results are illustrated by some numerical experiments.