Analysis of the Fourier series Dirichlet-to-Neumann boundary condition of the Helmholtz equation and its application to finite element methods

TL;DR

This paper analyzes the Fourier series Dirichlet-to-Neumann boundary condition for the Helmholtz equation, establishing well-posedness and error estimates in finite element methods for elastic and acoustic wave problems.

Contribution

It provides a detailed analysis of the DtN boundary condition's effect on finite element solutions for Helmholtz problems, including well-posedness and error bounds.

Findings

Established well-posedness of the variational problem with DtN boundary condition.

Derived a priori error estimates considering discretization and truncation effects.

Numerical results confirm the scheme's accuracy and effectiveness.

Abstract

It is well known that the Fourier series Dirichlet-to-Neumann (DtN) boundary condition can be used to solve the Helmholtz equation in unbounded domains. In this work, applying such DtN boundary condition and using the finite element method, we solve and analyze a two dimensional transmission problem describing elastic waves inside a bounded and closed elastic obstacle and acoustic waves outside it. We are mainly interested in analyzing the DtN boundary condition of the Helmholtz equation in order to establish the well-posedness results of the approximated variational equation, and further derive a priori error estimates involving effects of both the finite element discretization and the DtN boundary condition truncation. Finally, some numerical results are presented to illustrate the accuracy of the numerical scheme.