Analysis of a Moore–Gibson–Thompson thermoelastic problem

Abstract

In this work, we numerically consider a thermoelastic problem where the thermal law is modeled using the so-called Moore–Gibson–Thompson equation. This thermomechanical problem is written as a coupled system composed of a hyperbolic partial differential equation for a transformation of the displacement field and a parabolic partial differential equation for a transformation of the temperature. Its variational formulation is written in terms of the derivatives of the above transformed functions, leading to a coupled linear system made of two first-order variational equations. Then, a fully discrete algorithm is introduced and a discrete stability property is proved. A priori error estimates are also provided, from which the linear convergence is derived under suitable regularity conditions. Finally, some numerical results are shown, including the numerical convergence of the approximations, comparisons with the Lord–Shulman and type III Green–Naghdi theories, and two-dimensional examples which demonstrate the behavior of the solution.