Boundary element methods for Dirichlet boundary control problems

In this article we discuss the application of boundary element methods for the solution of Dirichlet boundary control problems subject to the Poisson equation with box constraints on the control. The solutions of both the primal and adjoint boundary value problems are given by representation formulae, where the state enters the adjoint problem as volume density. To avoid the related volume potential we apply integration by parts to the representation formula of the adjoint problem. This results in a system of boundary integral equations which is related to the Bi-Laplacian. For the related Dirichlet to Neumann map, we analyse two different boundary integral representations. The first one is based on the use of single and double layer potentials only, but requires some additional assumptions to ensure stability of the discrete scheme. As a second approach, we consider the symmetric formulation which is based on the use of the Calderon projector and which is stable for standard boundary element discretizations. For both methods, we prove stability and related error estimates which are confirmed by numerical examples.