An energy space finite element approach for elliptic Dirichlet boundary control problems

Abstract

In this paper we present a finite element analysis for a Dirichlet boundary control problem where the Dirichlet control is considered in a convex closed subspace of the energy space H 1/2 (Γ). As an equivalent norm in H 1/2 (Γ) we use the energy norm induced by the so-called Steklov-Poincaré operator which realizes the Dirichlet to Neumann map, and which can be implemented by using standard finite element methods. The presented stability and error analysis of the discretization of the resulting variational inequality is based on the mapping properties of the solution operators related to the primal and adjoint boundary value problems, and their finite element approximations. Some numerical results are given, which confirm on one hand the theoretical estimates, but on the other hand indicate the differences when modelling the control in L 2 (Γ).

Error estimates for variational normal derivatives and Dirichlet control problems with energy regularization

“…where z and z h are the continuous and discrete optimal control. This confirms the behavior figured out in the numerical experiments from [23] on the unit square, where the rate 3/2 was predicted numerically. The conjecture that this rate is achieved on arbitrary convex polygonal domains is obviously wrong.…”

“…al. [23]. It has to be noted that the behavior near the corners is in this approach just shifted to the tangential derivatives of the control.…”

“…Note, that the optimal state is in both approaches equivalent. Error estimates for approximate solutions of the Dirichlet control problem are discussed already in [23] where all variables are approximated by piecewise linear finite elements. For this approach, and in case of convex computational domains, the convergence rate of 1 for the control in the H 1/2 (Γ)-norm was proved, but in the numerical experiments a higher convergence rate is observed.…”

“…Such an error term appears due do the approximation of the Steklov-Poincaré operator z → ∂ n u(z) used to realize the H 1/2 (Γ)-norm, and the approximation for the normal derivative of the adjoint state variable which appears in the optimality condition. A worst-case estimate for variational normal derivatives in the H −1/2 (Γ)-norm, as used in [23], can be easily derived when using a trace theorem and standard finite element error estimates. Sharp error estimates require some more effort and will be discussed intensively in the present article.…”

Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations

“…However, the realization of the H 1/2 (Γ C )-inner product would introduce undesirable computational complexities, see e.g. [37] for further details. Several alternatives have been proposed to overcome this issue, for instance:…”