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.

KEYWORDS:

• Mixtures
• viscoelasticity
• finite element approximations
• error estimates
• numerical simulations

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 74E30
• 74D05
• 65M60
• 65M15
• 65M12

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of J. R. Fernández has been supported by the Ministerio de Economía y Competitividad under the research project MTM2015-66640-P (with the participation of FEDER). The work of A. Magaña and R. Quintanilla is partially supported by  projects Análisis Matemático de Problemas de la Termomecánica [MTM2016-74934-P],  [AEI/FEDER, UE] of the Spanish Ministry of Economy and Competitiveness and Análisis Matemático Aplicado a la Termomecánica submitted to the  Spanish Ministry of Science, Innovation and Universities.