In this paper we study, from the numerical point of view, a thermoelastic problem with dual-phase-lag heat conduction. The variational formulation is written as a coupled system of hyperbolic linear variational equations. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A discrete stability result is proved and a priori error estimates are obtained, from which the linear convergence of the algorithm is deduced under suitable additional regularity conditions. Finally, some two-dimensional numerical simulations are presented to demonstrate the accuracy of the approximation and the behaviour of the solution.

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