Numerical Analysis for a Nonlinear Coupled Thermoelastic System with Acoustic Boundary Conditions

Abstract

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 $\theta$, and the normal displacement to the boundary $\delta$, with nonlinearities $|u{|}^{d}u$ in the domain and $g(\cdot ,u^{\prime })$ on the boundary, in addition to a nonlocal term $\beta ({\int }_{\Omega }\theta dx)\Delta \theta$ 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 $r⩾1$ 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 $\mathcal{O}(h^{r+\frac{1}{2}}+{\tau }^2)$. A bootstrap argument is employed to ensure the uniform boundedness of $\Vert U_h^n{\Vert }_{L^{\infty }}$, 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 $L^{\infty }$ regularity.