On the Schur complement form of the Dirichlet-to-Neumann operator

The Dirichlet-to-Neumann operator relates the values of the normal derivative of a scalar field to the values of the field itself on the boundary of a simply connected domain. Although it is easy to prove analytically that the Dirichlet-to-Neumann operator is self-adjoint, the discretization of the pertinent Green’s integral equation by means of the usual Galerkin approach generally results in a non-symmetric matrix representation of that operator. In this paper we remedy this by means of a Hamiltonian Schur complement method and compare it with other symmetrization approaches. It is also shown that the problem of E-wave scattering by a lossy dielectric cylinder can be nicely simplified by means of the Dirichlet-to-Neumann operator. A numerical Galerkin implementation demonstrates the strength and versatility of the present approach.