Stability analysis for acoustic waveguides. Part 3: impedance boundary conditions

Leszek Demkowicz; Jay Gopalakrishnan; Norbert Heuer

Applications of Mathematics (2024)

  • Volume: 69, Issue: 5, page 633-651
  • ISSN: 0862-7940

Abstract

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A model two-dimensional acoustic waveguide with lateral impedance boundary conditions (and outgoing boundary conditions at the waveguide outlet) is considered. The governing operator is proved to be bounded below with a stability constant inversely proportional to the length of the waveguide. The presence of impedance boundary conditions leads to a non self-adjoint operator which considerably complicates the analysis. The goal of this paper is to elucidate these complications and tools that are applicable, as simply as possible. This work is a continuation of prior waveguide studies (where self-adjoint operators arose) by J. M. Melenk et al. (2023), and L. Demkowicz et al. (2024).

How to cite

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Demkowicz, Leszek, Gopalakrishnan, Jay, and Heuer, Norbert. "Stability analysis for acoustic waveguides. Part 3: impedance boundary conditions." Applications of Mathematics 69.5 (2024): 633-651. <http://eudml.org/doc/299318>.

@article{Demkowicz2024,
abstract = {A model two-dimensional acoustic waveguide with lateral impedance boundary conditions (and outgoing boundary conditions at the waveguide outlet) is considered. The governing operator is proved to be bounded below with a stability constant inversely proportional to the length of the waveguide. The presence of impedance boundary conditions leads to a non self-adjoint operator which considerably complicates the analysis. The goal of this paper is to elucidate these complications and tools that are applicable, as simply as possible. This work is a continuation of prior waveguide studies (where self-adjoint operators arose) by J. M. Melenk et al. (2023), and L. Demkowicz et al. (2024).},
author = {Demkowicz, Leszek, Gopalakrishnan, Jay, Heuer, Norbert},
journal = {Applications of Mathematics},
keywords = {acoustic waveguides; well-posedness analysis},
language = {eng},
number = {5},
pages = {633-651},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Stability analysis for acoustic waveguides. Part 3: impedance boundary conditions},
url = {http://eudml.org/doc/299318},
volume = {69},
year = {2024},
}

TY - JOUR
AU - Demkowicz, Leszek
AU - Gopalakrishnan, Jay
AU - Heuer, Norbert
TI - Stability analysis for acoustic waveguides. Part 3: impedance boundary conditions
JO - Applications of Mathematics
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 5
SP - 633
EP - 651
AB - A model two-dimensional acoustic waveguide with lateral impedance boundary conditions (and outgoing boundary conditions at the waveguide outlet) is considered. The governing operator is proved to be bounded below with a stability constant inversely proportional to the length of the waveguide. The presence of impedance boundary conditions leads to a non self-adjoint operator which considerably complicates the analysis. The goal of this paper is to elucidate these complications and tools that are applicable, as simply as possible. This work is a continuation of prior waveguide studies (where self-adjoint operators arose) by J. M. Melenk et al. (2023), and L. Demkowicz et al. (2024).
LA - eng
KW - acoustic waveguides; well-posedness analysis
UR - http://eudml.org/doc/299318
ER -

References

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  1. Bari, N. K., Biorthogonal systems and bases in Hilbert space, Moskov. Gos. Univ. Učenye Zapiski Matematika 148/4 (1951), 69-107 Russian. (1951) MR0050171
  2. Demkowicz, L., Gopalakrishnan, J., 10.1002/num.20640, Numer. Methods Partial Differ. Equations 27 (2011), 70-105. (2011) Zbl1208.65164MR2743600DOI10.1002/num.20640
  3. Demkowicz, L., Gopalakrishnan, J., Muga, I., Zitelli, J., 10.1016/j.cma.2011.11.024, Comput. Methods Appl. Mech. Eng. 213-216 (2012), 126-138. (2012) Zbl1243.76059MR2880509DOI10.1016/j.cma.2011.11.024
  4. Demkowicz, L., Melenk, J. M., Badger, J., Henneking, S., 10.1007/s10444-024-10130-x, Adv. Comput. Math. 50 (2024), Article ID 35, 32 pages. (2024) Zbl07845580MR4739882DOI10.1007/s10444-024-10130-x
  5. Glazman, I. M., On expansibility in a system of eigenelements of dissipative operators, Usp. Mat. Nauk 13 (1958), 179-181 Russian. (1958) Zbl0081.12204MR0097726
  6. Gohberg, I. C., Krein, M. G., 10.1090/mmono/018, Translations of Mathematical Monographs 18. AMS, Providence (1969). (1969) Zbl0181.13503MR0246142DOI10.1090/mmono/018
  7. Halla, M., 10.1137/22M1509266, SIAM J. Math. Anal. 55 (2023), 5445-5463. (2023) Zbl1527.35190MR4649397DOI10.1137/22M1509266
  8. Keldyš, M. V., On the characteristic values and characteristic functions of certain classes of non-self-adjoint equations, Dokl. Akad. Nauk SSSR, N. Ser. 77 (1951), 11-14 Russian. (1951) Zbl0045.39402MR0041353
  9. Melenk, J. M., Demkowicz, L., Henneking, S., 10.48550/arXiv.2307.04521, Available at https://arxiv.org/abs/2307.04521 (2023), 45 pages. (2023) MR4739882DOI10.48550/arXiv.2307.04521
  10. Schwartz, J. T., 10.2140/pjm.1954.4.415, Pac. J. Math. 4 (1954), 415-458. (1954) Zbl0056.34901MR0063568DOI10.2140/pjm.1954.4.415

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