Stability and convergence of modal approximations in coupled thermoelastic systems: Theory and simulation

Imane Essadeq; Salem Nafiri; Saad Benjelloun; Anoir Essadat Fettouh

doi:10.3934/dcdss.2026070

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2026. Doi: 10.3934/dcdss.2026070

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Stability and convergence of modal approximations in coupled thermoelastic systems: Theory and simulation

1.
LAGES Laboratory, MoNum Team, Hassania School of Public Works, Casablanca, Morocco
2.
DeVinci Higher Education, De Vinci Research Center, Paris, France

^*Corresponding author: Imane Essadeq
^*Corresponding author: Imane Essadeq

Dedicated to the memory of Professor Hammadi Bouslous

Received: September 14, 2025

Revised: January 16, 2026

Early access: February 27, 2026

Published online: February 27, 2026

Abstract / Introduction Full Text(HTML) Figure(7) / Table(2) Related Papers Cited by

Abstract

In this work, we review and analyze both the theoretical and numerical aspects of strongly and weakly coupled thermoelastic systems. By employing spectral analysis techniques and establishing uniform resolvent estimates, we derive uniform polynomial decay rates for the associated semigroups under a suitable class of boundary conditions. Particular attention is paid to the role of modal approximations in energy analysis. The theoretical results are complemented by numerical experiments that illustrate how the regularity of initial data, smooth versus nonsmooth, affects the observed decay rates, providing deeper insight into the interplay between spectral structure and energy dissipation.

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Mathematics Subject Classification: Primary: 35L55, 35K05, 35Q74, 65M60; Secondary: 35B40, 35M33, 65M12, 93D20.

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Figure 1. Eigenvalues of $ \mathcal{S}_{n} $

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Figure 2. Eigenvalues of $ \mathcal{W}_{n} $

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Figure 3. Uniform decay of the energy

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Figure 4. No effect of initial data smoothness on the exponential decay of $ E_{s,n}(t) $

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Figure 5. Effect of initial data smoothness on the polynomial decay of $ E_{w,n}(t) $

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Figure 6. Plot of the discontinuous velocity $ u_t(x,0) $

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Figure 7. Effect of initial data discontinuities on the polynomial decay of energy

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Table 1. Notation for coefficients and variables in the thermoelastic system

Symbol	Description	Units
$ u $	Displacement field (along the domain)	meters (m)
$ \theta $	Absolute temperature field	kelvin (K)
$ \partial_x u $	Strain	Dimensionless
$ \partial_x\partial_t u $	Rate of change of strain	$ 1/\text{s} $
$ \partial_t u $	Velocity	m/s
$ \partial^2_t u $	Acceleration	m/s²
$ \partial_x \theta $	Temperature gradient (heat flux)	K/m
$ \partial_t \theta $	Rate of change of temperature	K/s
$ \partial^2_x \theta $	Heat diffusion term (in Fourier's law)	K/m²
$ \theta_0 $	Reference (initial) temperature	kelvin (K)
$ \rho $	Mass density of the material	kg/m³
$ h_0 $	Specific heat at constant strain	J/(kg$ \cdot $K)
$ k $	Thermal conductivity	W/(m$ \cdot $K)
$ \lambda $	First Lamé constant	pascal (Pa)
$ \mu $	Second Lamé constant (shear modulus)	pascal (Pa)
$ \alpha $	Linear thermal expansion coefficient	1/K
$ L $	Length of the domain	meters (m)
$ t $	Time variable	seconds (s)
$ x $	Spatial variable	meters (m)
Derived quantities
$ \gamma = \left( \frac{\theta_0 \alpha^2 (3\lambda + 2\mu)^2}{h_0(\lambda + 2\mu)} \right)^{1/2} $	Coupling strength in the scaled system (very small in comparison to 1)	dimensionless
$ c^2 = \frac{h_0^2(\lambda + 2\mu)L^2}{k^2\rho} $	Normalized wave speed squared (often scaled to 1)	dimensionless

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Table 2. Distance between $ \sigma(\mathcal{S}_{n}) $ and the imaginary axis for the spectral element method

$ n $	$ \min\{-\text{Re}(\lambda), \lambda\in\sigma(\mathcal{S}_{n})\} $
8	$ 8.9227\times 10^{-4} $
16	$ 8.9383\times 10^{-4} $
24	$ 8.9402\times 10^{-4} $
32	$ 8.9407\times 10^{-4} $

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