Numerical analysis of a viscoelastic mixture problem

In this paper, we study, from the numerical point of view, a mixture problem involving a viscoelastic material and an elastic one. The mechanical problem is written as a linear system of two coupled hyperbolic partial differential equations. An existence and uniqueness result and an energy decay property are recalled. Then, fully discrete approximations are introduced by using the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. A priori error estimates are proved from which, under suitable regularity conditions, the linear convergence of the algorithm is derived. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximations and the behaviour of the solutions.