Abstract
AbstractIn 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. Copyright © 2010 John Wiley & Sons, Ltd.
Highlights
Optimal control problems of elliptic or parabolic partial differential equations with a Dirichlet boundary control play an important role, for example, in the context of computational fluid mechanics, see, e.g., [9, 11, 12], and the references given therein
While the first approach is based on the use of the standard boundary integral equation based on the Bi–Laplace fundamental solution, the additional use of the normal derivative of the related representation formula results in a symmetric formulation, which is symmetric in the discrete case
The most popular choice is to consider L2(Γ) as control space. This choice allows the use of a piecewise constant control function, the associated partial differential equation has to be considered within an ultra–weak variational formulation, see, for example, [20], and [2] for an appropriate finite element approximation using standard piecewise linear basis functions
Summary
Optimal control problems of elliptic or parabolic partial differential equations with a Dirichlet boundary control play an important role, for example, in the context of computational fluid mechanics, see, e.g., [9, 11, 12], and the references given therein. The use of the ultra–weak variational formulation of the primal Dirichlet boundary value problem in the context of an optimal control problem requires the adjoint variable p to be sufficiently regular, i.e., p ∈ H2(Ω) ∩ H01(Ω).
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