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).
The solution of eigenvalue problems for partial differential operators by using boundary integral equation methods usually involves some Newton potentials which may be resolved by using a multiple reciprocity approach. Here we propose an alternative approach which is in some sense equivalent to the above. Instead of a linear eigenvalue problem for the partial differential operator we consider a nonlinear eigenvalue problem for an associated boundary integral operator. This nonlinear eigenvalue problem can be solved by using some approriate iterative scheme, here we will consider a Newton scheme. We will discuss the convergence and the boundary element discretization of this algorithm, and give some numerical results.
In this paper we analyse constrained optimal Dirichlet boundary control problems subject to the linear heat equation. We propose to use boundary integral equations to solve the coupled optimality system, and we present results on unique solvability and related a priori error estimates for a symmetric Galerkin boundary element method. A numerical example confirms the analytical results.
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