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

Xu, Liwei and Yin, Tao (2021) Analysis of the Fourier series Dirichlet-to-Neumann boundary condition of the Helmholtz equation and its application to finite element methods. Numerische Mathematik, 147 (4). pp. 967-996. ISSN 0029-599X. doi:10.1007/s00211-021-01195-7. https://resolver.caltech.edu/CaltechAUTHORS:20210325-155505378

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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 analyze and solve 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 truncation of DtN map. Finally, some numerical results are presented to illustrate the accuracy of the numerical scheme.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/s00211-021-01195-7DOIArticle
https://rdcu.be/ci70wPublisherFree ReadCube access
https://arxiv.org/abs/1609.00583arXivDiscussion Paper
Alternate Title:A priori error estimates of the DtN-FEM: fluid-solid interaction problems
Additional Information:© 2021 The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature. Received 07 August 2018; Revised 20 January 2021; Accepted 09 March 2021; Published 25 March 2021. The work of L. Xu is partially supported by a Key Project of the Major Research Plan of NSFC (No. 91630205), and NSFC Grant (11771068, 12071060), and he also would like to thank Prof. G.C. Hsiao and Prof. J.E. Pasciak for their invaluable encouragements and suggestions which are of great importance for the completion of this work.
Funders:
Funding AgencyGrant Number
National Natural Science Foundation of China91630205
National Natural Science Foundation of China11771068
National Natural Science Foundation of China12071060
Issue or Number:4
Classification Code:Mathematics Subject Classification: 35J25; 65N30; 74F10
DOI:10.1007/s00211-021-01195-7
Record Number:CaltechAUTHORS:20210325-155505378
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210325-155505378
Official Citation:Xu, L., Yin, T. Analysis of the Fourier series Dirichlet-to-Neumann boundary condition of the Helmholtz equation and its application to finite element methods. Numer. Math. 147, 967–996 (2021). https://doi.org/10.1007/s00211-021-01195-7
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108564
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:25 Mar 2021 23:04
Last Modified:22 Apr 2021 18:38

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