An a priori error analysis of the porous thermoelastic Coleman–Gurtin model

In this work, we study, from the numerical point of view, a poro-thermoelastic problem where the heat conduction is modeled by using the Coleman–Gurtin law. This is written as a linear system of partial differential equations written in terms of the displacements, the porosity (or volume fraction), and the temperature. Then, we introduce a fully discrete approximation of a weak form of the thermomechanical problem, based on the classical finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. We prove a discrete stability property and a main a priori error estimates result, which allows us to conclude the linear convergence of the approximations under suitable additional regularity. Finally, we present some numerical simulations to demonstrate the convergence and the decay of the discrete energy.