Numerical behavior for quasi static thermoelasticity without positive definite elasticity

This paper presents a numerical study of the energetic behavior of some quasi-static thermoelastic problems in one- and two-dimensional settings. Firstly, we describe the two-dimensional thermoelastic problem decomposing the elastic tensor into two parts: the first one is positively defined for the first component of the displacement field, and the second one is negatively defined for the second component. The variational formulation is also derived. Restricting ourselves to the one-dimensional setting and assuming that the elastic coefficient is negative, we prove that the exponential energy decay follows if the coupling coefficient is smaller than the square root of the product between the heat capacity and the elastic coefficient. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. Some numerical simulations are performed: in a first onedimensional example, we show the decay of the discrete energy depending on the value of the coupling coefficient and the heat diffusion. Secondly, two dimensional studies are considered depending on the expression of the elastic tensors, including diagonal matrices with the same eigenvalue, diagonal matrices with different eigenvalues and full matrices.