Fully Discrete Approximations and an a Priori Error Analysis of a Two–Temperature Thermo–Elastic Model with Microtemperatures

In this paper, we consider, from a numerical point of view, a two-temperature poro-thermoelastic problem. The model is written as a coupled linear system of hyperbolic and elliptic partial differential equations. An existence result is proved and energy decay properties are recalled. Then we introduce a fully discrete approximation by using the finite element method and the implicit Euler scheme. Some a priori error estimates are obtained, from which the linear convergence of the approximation is deduced under an appropriate additional regularity. Finally, some numerical simulations are performed to demonstrate the accuracy of the approximation, the decay of the discrete energy and the behaviour of the solution depending on a constitutive parameter.