Analysis and approximation of frictionless contact problems between two piezoelectric bodies with adhesion

We consider a mathematical frictionless contact problem between two electro-elastic bodies. The contact is modelled with normal compliance and adhesion. We provide a variational formulation for the problem and prove the existence of a unique weak solution. The proofs are based on arguments of time-dependent variational inequalities, the Cauchy-Lipschitz Theorem and the Banach Fixed-Point Theorem. Then, a discrete scheme is introduced based on the finite element method to approximate the spatial variable. Furthermore, we provide optimal a priori error estimates for the displacements, the electric potential and the bonding at the contact interface.