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Acoustic vibration problem for dissipative fluids


Authors: Felipe Lepe, Salim Meddahi, David Mora and Rodolfo Rodríguez
Journal: Math. Comp. 88 (2019), 45-71
MSC (2010): Primary 65N25, 65N30, 76M10
DOI: https://doi.org/10.1090/mcom/3336
Published electronically: March 26, 2018
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Abstract: In this paper we analyze a finite element method for solving a quadratic eigenvalue problem derived from the acoustic vibration problem for a heterogeneous dissipative fluid. The problem is shown to be equivalent to the spectral problem for a non-compact operator and a thorough spectral characterization is given. The numerical discretization of the problem is based on Raviart-Thomas finite elements. The method is proved to be free of spurious modes and to converge with optimal order. Finally, we report numerical tests which allow us to assess the performance of the method.


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Additional Information

Felipe Lepe
Affiliation: CI$^{\mathrm{2}}$MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Email: flepe@ing-mat.udec.cl

Salim Meddahi
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s/n, Oviedo, Spain
Email: salim@uniovi.es

David Mora
Affiliation: Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile; and Centro de Investigación en Ingeniería Matemática (CI$^{\mathrm{2}}$MA), Universidad de Concepción, Concepción, Chile
Email: dmora@ubiobio.cl

Rodolfo Rodríguez
Affiliation: CI$^{\mathrm{2}}$MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Email: rodolfo@ing-mat.udec.cl

DOI: https://doi.org/10.1090/mcom/3336
Keywords: Spectral problems, dissipative fluid, finite elements, error estimates.
Received by editor(s): October 10, 2016
Received by editor(s) in revised form: July 24, 2017, and October 11, 2017
Published electronically: March 26, 2018
Additional Notes: The first author was supported by a CONICYT fellowship (Chile).
The second author was supported by Spain’s Ministry of Economy Project MTM2013-43671-P
The third author was partially supported by CONICYT-Chile through FONDECYT project 1140791 (Chile) and by DIUBB through project 151408 GI/VC, Universidad del Bío-Bío (Chile)
The fourth author was partially supported by BASAL project CMM, Universidad de Chile (Chile).
Article copyright: © Copyright 2018 American Mathematical Society

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