Analysis of two thermoelastic problems with the Green–Lindsay model

In this paper, we analyze, from the numerical point of view, two thermo-elastic problems involving the Green–Lindsay theory. The coupling term is different for each case, involving second order or first order spatial derivatives, respectively. The variational formulation leads to a linear coupled system which is written in terms of the velocity and temperature speed. An existence and uniqueness results and the exponential energy decay for the problem with the stronger coupling are recalled. The polynomial energy decay for the weaker coupling is then proved but using the theory of linear semigroups. Then, a fully discrete approximation is introduced using the finite element method and an implicit scheme. A discrete stability property and a main a priori error estimates result are shown, from which we can derive the linear convergence of the approximations. Finally, some numerical simulations are presented to demonstrate the accuracy of the algorithm, the discrete energy decay and the dependence on the relaxation parameter.