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Boundary element methods for Dirichlet boundary control problems
Author(s) -
Of Günther,
Phan Thanh Xuan,
Steinbach Olaf
Publication year - 2010
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1356
Subject(s) - mathematics , robin boundary condition , mixed boundary condition , neumann boundary condition , boundary value problem , boundary (topology) , mathematical analysis , boundary conditions in cfd , boundary element method , singular boundary method , free boundary problem , dirichlet boundary condition , dirichlet distribution , boundary knot method , finite element method , physics , thermodynamics
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. Copyright © 2010 John Wiley & Sons, Ltd.