Dirichlet-to-Neumann and Neumann-to-Dirichlet methods for bound states of the Helmholtz equation

TL;DR

This paper introduces Dirichlet-to-Neumann and Neumann-to-Dirichlet surface integral operator methods for computing bound states of the Helmholtz equation, adapting techniques from quantum mechanics and employing a variational principle.

Contribution

It develops novel surface integral operator methods for Helmholtz bound states, extending DtN and NtD techniques from Schrödinger equations with a new variational framework.

Findings

Methods successfully compute Helmholtz bound states.

Variational principle allows discontinuous trial functions.

Numerical example demonstrates method effectiveness.

Abstract

Two methods for computing bound states of the Helmholtz equation in a finite 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 Schrodinger equation in R^3. 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.