A spectral decomposition method to approximate DtN maps in complicated waveguides

TL;DR

This paper introduces a spectral decomposition approach to efficiently approximate Dirichlet-to-Neumann maps in complex waveguides, enabling accurate wave simulation in bounded domains with exponential convergence guarantees.

Contribution

The paper presents a novel spectral decomposition method for approximating DtN maps in complicated waveguides, improving computational efficiency and accuracy over existing techniques.

Findings

Exponential decay of generalized eigenfunctions enables finite truncation of DtN map.

The proposed method achieves exponential convergence in approximation.

Numerical examples demonstrate the efficiency and accuracy of the approach.

Abstract

In this paper, we propose a new spectral decomposition method to simulate waves propagating in complicated waveguides. For the numerical solutions of waveguide scattering problems, an important task is to approximate the Dirichlet-to-Neumann map efficiently. From previous results, the physical solution can be decomposed into a family of generalized eigenfunctions, thus we can write the Dirichlet-to-Neumann map explicitly by these functions. From the exponential decay of the generalized eigenfunctions, we approximate the Dirichlet-to-Neumann (DtN) map by a finite truncation and the approximation is proved to converge exponentially. With the help of the truncated DtN map, the unbounded domain is truncated into a bounded one, and a variational formulation for the problem is set up in this bounded domain. The truncated problem is then solved by a finite element method. The error estimation is also provided for the numerical algorithm and numerical examples are shown to illustrate the efficiency of the algorithm.