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Stability and Convergence of Modal Approximations in Coupled Thermoelastic Systems: Theory and Simulation

I. Essadeq, S. Nafiri, S. Benjelloun, A. E. Fettouh

TL;DR

The paper analyzes stability and convergence for strongly and weakly coupled thermoelastic systems using semigroup theory, spectral analysis, and uniform resolvent estimates. It shows that strongly coupled models exhibit uniform exponential decay, and establishes this property for their modal approximations under multiple boundary conditions. For weakly coupled models, exponential decay is impossible, but uniform polynomial decay is proven both in the continuous setting and for modal discretizations, with an explicit rate in the discretized schemes (α=2). Numerical experiments corroborate the theory, revealing that energy decay in the weakly coupled case is highly sensitive to initial-data regularity and boundary conditions. Overall, the work provides a rigorous framework ensuring that modal approximations preserve the long-time dissipation characteristics of the continuous thermoelastic systems, with practical implications for reliable long-time simulations and control design.

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.

Stability and Convergence of Modal Approximations in Coupled Thermoelastic Systems: Theory and Simulation

TL;DR

The paper analyzes stability and convergence for strongly and weakly coupled thermoelastic systems using semigroup theory, spectral analysis, and uniform resolvent estimates. It shows that strongly coupled models exhibit uniform exponential decay, and establishes this property for their modal approximations under multiple boundary conditions. For weakly coupled models, exponential decay is impossible, but uniform polynomial decay is proven both in the continuous setting and for modal discretizations, with an explicit rate in the discretized schemes (α=2). Numerical experiments corroborate the theory, revealing that energy decay in the weakly coupled case is highly sensitive to initial-data regularity and boundary conditions. Overall, the work provides a rigorous framework ensuring that modal approximations preserve the long-time dissipation characteristics of the continuous thermoelastic systems, with practical implications for reliable long-time simulations and control design.

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.
Paper Structure (35 sections, 16 theorems, 163 equations, 7 figures, 2 tables)

This paper contains 35 sections, 16 theorems, 163 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Mathematical Formulation and State of the Art
  4. Problem Statement and Contributions
  5. Preliminaries and Framework Setting
  6. Notations and Functional Spaces
  7. Semigroup Formulation
  8. Energy Functional and Stability Definitions
  9. Approximation Strategy
  10. Uniform Exponential Stability of the Strongly Coupled Thermoelastic System $(S_s)$
  11. Spectral Analysis of System ($S_s$)
  12. Exponential Stability of System $(S_s)$
  13. Uniform Exponential Stability of Approximation of $(S_s)$
  14. Approximation of System $(S_s)$ under Dirichlet--Dirichlet (BCs)
  15. Approximation of System $(S_s)$ under Dirichlet--Neumann (BCs)
  16. ...and 20 more sections

Key Result

Theorem 3.1

Asymptotically, there are at most two branches for the eigenvalues of operator $\mathcal{S}$. One branch is along the negative axis and the other branch approaches a vertical line $\Re\lambda = -\frac{\gamma^2}{2}$ parallel to the imaginary axis.

Figures (7)

  • Figure 1: Eigenvalues of $\mathcal{S}_{n}$.
  • Figure 2: Eigenvalues of $\mathcal{W}_{n}$.
  • Figure 3: Uniform decay of the energy.
  • Figure 4: No effect of initial data smoothness on the exponential decay of $E_{s,n}(t)$.
  • Figure 5: Effect of initial data smoothness on the polynomial decay of $E_{w,n}(t)$.
  • ...and 2 more figures

Theorems & Definitions (28)

  • Definition 2.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1: guo1997asymptotic
  • Theorem 3.2: liu1994uniform, Corollary 2.3
  • Theorem 3.3: liu1994uniform, Theorem 3.1
  • Theorem 3.4: liu1994uniform Theorem 3.2
  • Proposition 3.1
  • Lemma 4.1
  • proof
  • ...and 18 more