Direct numerical identification of boundary values in the Laplace equation

An inverse boundary value problem for the Laplace equation is considered. The Dirichlet and the Neumann data are prescribed on respective part of the boundary, while there is the second part of the boundary where no boundary data are given. There is the third part of the boundary where the Robin condition is prescribed. This ill-posed problem of finding unknown values along the whole boundary is reformulated in terms of the variational problem, which is then recast into primary and adjoint boundary value problems of the Laplace equation in conventional forms. A direct method for numerical solution of the boundary value problems using the boundary element method is presented.