Analysis of contact problems of porous thermoelastic solids

In this article, we study a contact problem between a one-dimensional porous thermoelastic layer and a rigid obstacle. The mechanical problem consists of a coupled system of two hyperbolic partial differential equations and a parabolic one. By defining penalized problems, an energy decay property is obtained. Then, fully discrete algorithms are introduced to approximate both penalized and Signorini problems using the finite element method and the implicit Euler scheme. Stability properties are shown for both problems and a priori error estimates are proved for the penalized problem, from which the linear convergence of the algorithm is derived. Finally, some numerical simulations are performed to demonstrate the accuracy of the approximation and the behavior of the solution.