Numerical analysis of a dual-phase-lag model with microtemperatures

Abstract In the last twenty years, the analysis of problems involving dual-phase-lag models has received an increasing attention. In this work, we consider the coupling between one of these models and the microtemperatures effects. In order to overcome the infinite speed paradox, two relaxation parameters are introduced for each evolution equation related to the temperature and the microtemperatures, leading to a system of linear hyperbolic partial differential equations. Its variational formulation is written in terms of the temperature acceleration and the microtemperatures acceleration. An energy decay property is proved. Next, fully discrete approximations are introduced by using the finite element method and the Euler scheme, proving a stability property and a discrete version of the energy decay, obtaining a priori error estimates and performing one- and two-dimensional numerical simulations.