Numerical approximation of a Bresse-Maxwell-Cattaneo system

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

In this paper, we analyze, from the numerical point of view, a new thermoelastic problem involving the so-called Bresse system. The heat conduction is modeled by using the Maxwell-Cattaneo law, which is of hyperbolic type. An existence and uniqueness result and an energy decay property are recalled. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. First, we prove that the discrete solution is stable, and secondly, we provide an a priori error analysis. This allows us to conclude the linear convergence under suitable additional regularity on the continuous solution. Finally, numerical results are presented to demonstrate the convergence of the scheme and the behavior of the discrete energy.

Keywords

Bresse-Maxwell-Cattaneo system / hyperbolic heat conduction / finite elements / discrete stability / error estimates / numerical results

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] J. A. C. Bresse, Cours de Mécanique Appliquée, Mallet Bachelier, Paris, 1859.

[2] J. E. Lagnese, G. Leugering, E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhauuser, Boston, 1994.

[3] T. F. Ma, R. Nunes, Singular limit and long-time dynamics of Bresse systems, SIAM J. Math. Anal., 49 (2017), 2468-2495. https://doi.org/10.1137/15M1039894

[4] S. Baibeche, L. Bouzetouta, F. Hebhoub, A. Lallouche, Well-posedness and energy decay for Bresse system with microtemperature effects in the presence of delay, SeMA J., 82 (2025), 137-155. https://doi.org/10.1007/s40324-024-00356-6

[5] H. Li, V. R. Cabanillas, B. Feng, Attractors of a thermoelastic Bresse system with mass diffusion, Z. Angew. Math. Mech., 104 (2024), e202300502. https://doi.org/10.1002/zamm.202300502

[6] J. E. Muñoz Rivera, M. G. Naso, Pointwise stabilization of Bresse systems, Z. Angew. Math. Phys., 74 (2023), 218. https://doi.org/10.1093/biosci/biae025

[7] M. I. M. Copetti, T. El Arwadi, J. R. Fernández, M. G. Naso, W. Youssef, Analysis of a contact problem for a viscoelastic Bresse system, ESAIM Math. Model. Num. Anal., 55 (2021), 887-911. https://doi.org/10.1051/m2an/2021015

[8] N. Bazarra, I. Bochicchio, J. R. Fernández, M. G. Naso, Thermoelastic Bresse system with dual-phase-lag, Z. Angew. Math. Phys., 72 (2021), 102. https://doi.org/10.1007/s00033-021-01536-4

[9] M. Afilal, A. Guesmia, A. Soufyane, New stability results for a linear thermoelastic Bresse system with second sound, Appl. Math. Optim., 83 (2021), 699-738. https://doi.org/10.1007/s00245-019-09560-7

[10] M. Afilal, A. Guesmia, A. Soufyane, Stability of a linear thermoelastic Bresse system with second sound under new conditions on the coefficients, Z. Angew. Math. Mech., 103 (2023), e202200275. https://doi.org/10.1002/zamm.202200275

[11] T. Suzuki, Y. Ueda, Lack of the strict dissipativity and modification for the dissipative Bresse system, J. Differ. Equations, 347 (2023), 24-55. https://doi.org/10.1016/j.jde.2022.11.030

[12] F. Alabau-Boussouira, J. E. Muñoz Rivera, D. S. Almeida Júnior, Stability to weak dissipative bresse system, J. Math. Anal. Appl., 374 (2011), 481-498. https://doi.org/10.1016/j.jmaa.2010.07.046

[13] D. S. Almeida Júnior, Conservative semidiscrete difference schemes for timoshenko systems, J. Appl. Math., 1 (2014), 686421. https://doi.org/10.1155/2014/686421

[14] J. Wang, X. Jiang, H. Zhang, X. Yang, A new fourth-order nonlinear difference scheme for the nonlinear fourth-order generalized burgers-type equation, J. Appl. Math. Comput., in press. https://doi.org/10.1007/s12190-025-02467-3

[15] X. Yang, L. Wu, H. Zhang, A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity, Appl. Math. Comput., 457 (2023), 128192. https://doi.org/10.1016/j.amc.2023.128192

[16] F. Dell’Oro, On the stability of Bresse and Timoshenko systems with hyperbolic heat conduction, J. Differ. Equations, 281 (2021), 148-198. https://doi.org/10.1016/j.jde.2021.02.009

[17] P. R. de Lima, H. D. Fernández Sare, General condition for exponential stability of thermoelastic Bresse systems with Cattaneo’s law, Commun. Pure Appl. Anal., 19 (2020), 3575-3596. https://doi.org/10.3934/cpaa.2020156

[18] C. Cattaneo, On a form of heat equation which eliminates the paradox of instantenous propagation, C. R. Acad. Sci. Paris, 247 (1958), 431-433.

[19] M. Renardy, W. J. Hrusa, J. A. Nohel, Mathematical Problems in Viscoelasticity, vol. 35 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific and Technical, New York, 1987.

[20] Z. Liu, B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys., 60 (2009), 54-69. https://doi.org/10.1007/s00033-008-6122-6

[21] P. G. Ciarlet, The finite element method for elliptic problems, in Handbook of Numerical Analysis. Vol. II, North-Holland, 1991, 17-352.

[22] M. Campo, J. R. Fernández, K. L. Kuttler, M. Shillor, J. M. Viaño, Numerical analysis and simulations of a dynamic frictionless contact problem with damage, Comput. Methods Appl. Mech. Eng., 196 (2006), 476-488. https://doi.org/10.1016/j.cma.2006.05.006