Dirichlet-to-Neumann and Neumann-to-Dirichlet methods for eigenvalues and eigenfunctions of the Laplace operator

Two domain decomposition methods for computing eigenvalues and eigenfunctions of the Laplace operator on a bounded domain are presented. The methods are formulated in terms of the Dirichlet-to-Neumann (DtN) and Neumann-to-Dirichlet (NtD) surface integral operators. They are adapted from the DtN and NtD methods for bound states of the Schrödinger equation in R3. A variational principle that enables the usage of the operators is constructed. The variational principle allows the use of discontinuous (in values or derivatives) trial functions. A numerical example presenting the usefulness of the DtN and NtD methods is given.