A problem with viscoelastic mixtures: numerical analysis and computational experiments

Abstract

In this paper, we study, from the numerical point of view, a dynamic problem involving a mixture of two viscoelastic solids. The mechanical problem is written as a system of two coupled partial differential equations. Its variational formulation is derived and an existence and uniqueness result, and an energy decay property, are recalled. Then, fully discrete approximations are introduced by using the classical finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are shown, from which we deduce the linear convergence of the algorithm. Finally, some numerical simulations, including examples in one and two dimensions, are presented to show the accuracy of the approximation and the behaviour of the solution.

Energy decay of a mixture with local viscoelasticity

“…Since the last two inner product terms converge to zero due to (26), and the real part of the first two inner product term vanishes, the real part of (31) gives…”

“…Several extensions to the other thermal theories can be found in previous works [28, 29]. It is worth recalling the recent papers concerning the analyticity of solutions in the case of isothermal viscoelasticity [30] and some numerical aspects concerning the same problem [31].…”

Decay of solutions for strain gradient mixtures

“…Several results concerning existence, uniqueness, continuous dependence and asymptotic stability can be found in the literature. See, for instance, [20][21][22][23][24][25][26][27] and [28].…”