A finite element method for elliptic Dirichlet boundary control problems
Provides a numerically feasible discretization for optimal Dirichlet boundary control problems, with rigorous error analysis, benefiting researchers in PDE-constrained optimization.
The paper develops a mixed finite element method for elliptic Dirichlet boundary control problems, avoiding direct computation of the H^{1/2} norm by using harmonic extension, and proves optimal error estimates for convex polygonal domains.
We consider the finite element discretization of an optimal Dirichlet boundary control problem for the Laplacian, where the control is considered in $H^{1/2}(Γ)$. To avoid computing the latter norm numerically, we realize it using the $H^{1}(Ω)$ 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.