Analysis of a problem arising in porous thermoelasticity of type II

In this article, we consider, from the numerical point of view, a linear thermo-porous-elastic model. The heat conduction is assumed to be of type II. The mechanical problem is written as a coupled system of three hyperbolic partial differential equations for the displacements, the porosity and the thermal displacements. Then, its variational formulation is derived, which is written as a coupled system of three hyperbolic linear variational equations in terms of the velocity, the porous speed and the temperature. An existence and uniqueness result, as well as an energy decay property, is recalled. The fully discrete approximation of the aforementioned problem is introduced by using the finite element method for the spatial approximation and the implicit Euler scheme to discretize the time derivatives. A stability property is proved, from which the energy decay of the discrete energy is deduced. Then, a priori error estimates are obtained, from which, under suitable regularity conditions, the linear convergence of the algorithm is derived. Finally, some numerical simulations are presented to show the accuracy of the approximation and the behavior of the solution.