Numerical Analysis for a Nonlinear Coupled Thermoelastic System with Acoustic Boundary Conditions
Abstract
This paper presents the first L2-norm error analysis for a nonlinear thermoelastic system with acoustic boundary conditions. The model involves three coupled variables: the acoustic potential u, the temperature θ, and the normal displacement to the boundary δ, with nonlinearities |u|d u in the domain and g(·,u') on the boundary, in addition to a nonlocal term β(∫Ωθdx)Δθ in the heat equation. We establish error estimates for the fully discrete problem, using the Crank-Nicolson method for time discretization and finite element method with basis of degree r geqslant 1 for space discretization. We demonstrate that the domain-boundary coupling induces a half-power loss in the spatial convergence order for the variable u, and this effect propagates to the remaining variables due to the thermoelastic and boundary couplings, resulting in the final error estimate 𝒪(hr+1/2 + τ2). A bootstrap argument is employed to ensure the uniform boundedness of |Uhn|L∞, which is essential for handling the nonlinearity. Numerical results using Lagrange finite element basis of degrees $ r = 1, 2, 3 $ (linear, quadratic, and cubic) corroborate the theoretical estimates and illustrate the transition from suboptimal to optimal convergence orders when the exact solution possesses additional L∞ regularity.
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BibTeX
@misc{alcantara2026numerical,
title = {{Numerical Analysis for a Nonlinear Coupled Thermoelastic System with Acoustic Boundary Conditions}},
author = {Adriano A. Alcântara and Bruno A. Carmo and Juan B. Límaco and Carlos M. Guzmán},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24704},
eprint = {2610.24704},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24704},
}
RIS
TY - UNPB
TI - Numerical Analysis for a Nonlinear Coupled Thermoelastic System with Acoustic Boundary Conditions
AU - Adriano A. Alcântara
AU - Bruno A. Carmo
AU - Juan B. Límaco
AU - Carlos M. Guzmán
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24704
UR - https://arxiv.org/abs/2610.24704
AB - This paper presents the first L^2-norm error analysis for a nonlinear thermoelastic system with acoustic boundary conditions. The model involves three coupled variables: the acoustic potential u, the temperature θ, and the normal displacement to the boundary δ, with nonlinearities |u|^d u in the domain and g(·,u') on the boundary, in addition to a nonlocal term β(∫_Ωθdx)Δθ in the heat equation. We establish error estimates for the fully discrete problem, using the Crank-Nicolson method for time discretization and finite element method with basis of degree r geqslant 1 for space discretization. We demonstrate that the domain-boundary coupling induces a half-power loss in the spatial convergence order for the variable u, and this effect propagates to the remaining variables due to the thermoelastic and boundary couplings, resulting in the final error estimate 𝒪(h^r+1/2 + τ^2). A bootstrap argument is employed to ensure the uniform boundedness of |U_h^n|_L^∞, which is essential for handling the nonlinearity. Numerical results using Lagrange finite element basis of degrees $ r = 1, 2, 3 $ (linear, quadratic, and cubic) corroborate the theoretical estimates and illustrate the transition from suboptimal to optimal convergence orders when the exact solution possesses additional L^∞ regularity.
ER -