Numerical Analysis Of A Thermoelastic Diffusion Problem With Voids

In this paper we consider, from the numerical point of view, a thermoelastic diffusion porous problem. This is written as a coupled system of two hyperbolic equations, for the displacement and porosity fields, and two parabolic equations, for the temperature and chemical potential fields. Its variational formulation leads to a coupled system of four parabolic variational equations in terms of the velocity, porosity speed, temperature and chemical potential. The existence and uniqueness of weak solutions, as well as an energy decay property, are recalled. Then, the numerical approximation 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 and some a priori error estimates are obtained, from which the convergence of the algorithm is derived and, under suitable additional regularity conditions, its linear convergence is de-duced. Finally, some numerical approximations are presented to demonstrate the accuracy of the algorithm and to show the behaviour of the solution.