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 $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) . As an equivalent norm in $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) we use the energy norm induced by the so-called Steklov---Poincare 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 $$L_2(\Gamma )$$ L 2 ( Γ ) .
Highlights
The focus is on the a priori error analysis of the finite element approximation to minimise the cost functional
The paper is organised as follows: In Sect. 2, we describe the considered Dirichlet boundary control problem, the primal boundary value problem, and the reduced cost functional as well as the related adjoint boundary value problem
To rewrite the Dirichlet boundary control problem (2.1)–(2.3) by using a reduced cost functional, we introduce a linear solution operator describing the application of the constraint (2.2)
Summary
The focus is on the a priori error analysis of the finite element approximation to minimise the cost functional. The aim of this paper is to present a numerical analysis of an energy space finite element approach when the control z is considered as an element of the boundary trace space H1/2(Γ), where an equivalent norm is induced by the so–called Steklov–Poincare operator which realizes the Dirichlet to Neumann map. Note that in this case the costs represent the energy of the harmonic extension of the Dirichlet control z. For an overview on the used Sobolev spaces in the domain and on the boundary, see, for example, [1, 31, 37, 40]
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