Numerical analysis of a problem involving an extensible thermoelastic beam with microtemperatures

Abstract In this article we study, from the numerical point of view, a problem involving an extensible thermoelastic beam with microtemperatures derived recently by Aouadi. The corresponding variational problem leads to a coupled system which is written in terms of the mechanical extensibility, the temperature, and the microtemperatures. An existence and uniqueness result is recalled. Then, we will focus on the numerical approximation of this problem. First, fully discrete approximations are introduced by using the finite element method for the spatial approximation and the implicit Euler scheme to discretize the time derivatives. Second, a discrete stability property is obtained, and an a priori error analysis is performed, from which the linear convergence is derived under adequate additional regularity conditions. Finally, some numerical simulations are presented in order to show the accuracy of the approximations and the behavior of the discrete energy under extensibility and microtemperatures effects.