A strain gradient problem with a fourth-order thermal law

In this paper, a strain gradient thermoelastic problem is studied from the numerical point of view. The heat conduction is modeled by using the type II thermal law and the second gradient of the thermal displacement is also included in the set of independent constitutive variables. An existence and uniqueness result is recalled. Then, the fully discrete approximations are introduced by using the implicit Euler scheme and the finite element method. A discrete stability property and a main a priori error estimates results are proved. Then, some numerical simulations are performed, including some issues as the numerical convergence of the approximations, the effect of two possible dissipative terms (second- and fourth-order) or a comparison with the type II strain gradient thermoelasticity.