Mathematical Analysis and Numerical Simulation of a Nonlinear Thermoelastic System

Abstract

In this paper, we give a theoretical and numerical analysis of a model for small vertical vibrations of an elastic membrane coupled with a heat equation and subject to linear mixed boundary conditions. We establish the existence, uniqueness, and a uniform decay rate for global solutions to this nonlinear non-local thermoelastic coupled system with linear boundary conditions. Furthermore, we introduced a numerical method based on finite element discretization in a spatial variable and finite difference scheme in time which results in a nonlinear system to be solved by Newton’s method. Numerical experiments for one-dimensional and two-dimensional cases are presented in order to estimate the rate of convergence of the numerical solution that confirm our analysis and show the efficiency of the method.

Keywords:

• Convergence order
• energy behavior
• existence and uniqueness
• numerical simulations
• thermoelastic coupled system

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 35L20
• 34B15
• 65M60

Additional information

Funding

The author M.A. Rincon acknowledges the partial support for research from CNPq of Brazil.