Numerical analysis of some dual-phase-lag models

In this paper we analyse, from the numerical point of view, two dual-phase-lag models appearing in the heat conduction theory. Both models are written as linear partial differential equations of third order in time. The variational formulations, written in terms of the thermal acceleration, lead to linear variational equations, for which existence and uniqueness results, and energy decay properties, are recalled. Then, fully discrete approximations are introduced for both models using the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. Discrete stability properties are proved, and a priori error estimates are obtained, from which the linear convergence of the approximations is derived. Finally, some numerical simulations are described in one and two dimensions to demonstrate the accuracy of the approximations and the behaviour of the solutions. (C) 2018 Elsevier Ltd. All rights reserved.