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Numerical analysis of a contact problem in poro‐thermoelasticity with microtemperatures

A dynamic contact problem, between a thermoelastic rod with voids and microtemperatures and a deformable obstacle, is numerically investigated in this work. The contact is modelled with a modification of the classical normal compliance contact condition. The mechanical problem consists of a coupled system of two hyperbolic partial differential equations and two parabolic ones. An existence and uniqueness result, and an energy decay property, are stated. Then, fully discrete approximations are introduced to approximate this nonlinear problem by using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, 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 behaviour of the solution.