Numerical analysis of a thermoelastic problem of Moore–Gibson–Thompson type with history dependence in the thermal displacement

In this work, we study, from the numerical point of view, a heat conduction model which is described by the history dependent version of the Moore–Gibson–Thompson equation. First, we consider the thermal problem, introducing a fully discrete approximation by means of the finite element method and the implicit Euler scheme. The discrete stability of its solution is proved, and an a priori error analysis is provided, which leads to the linear convergence imposing suitable regularity conditions. Secondly, we deal with the natural extension to the thermoelastic case. Following the analysis of the thermal problem, similar results are shown. Finally, we present some one-dimensional numerical simulations for both problems which demonstrate the accuracy of the approximations and the behavior of the discrete energies and the solutions.