Determining the response to the forces applied to an elastic solid containing an ideal fluid with constant density is essential in the engineering and biomedical fields. Therefore this paper aims to present and analyze a mixed finite element method for an interaction problem solid-fluid that contributes to understanding these areas. It is assumed transmission conditions are maintained at the fluid boundary and are given by the balance of forces and the equality of normal displacements. The mixed variational formulation that avoids the locking phenomenon, for the coupled problem is in terms of displacement, stress tensor, and rotation in the solid and by pressure and scalar potential in the fluid, the main contribution of this work. The first transmission condition is imposed in the definition of the space and the rest of the conditions appear naturally, which means Lagrange multipliers are not needed at the coupling border. The unknowns for the fluid and the solid are approximated by finite element subspaces of Lagrange and Arnold-Falk-Winther of order 1, which lead to a Galerkin scheme for the continuous problem. Also, the resulting Galerkin scheme is convergent and derives optimal convergence rates. Finally, the model is illustrated using a numerical example.

Physics is a science that has developed valuable contributions to the understanding of different phenomena; in particular, the concept of a dynamic system originates in Newtonian mechanics; a concept that allows predictions of phenomena that are changing over time. The evolution rule is an implicit relation that can be given by a differential equation; our goal is to use dynamic systems in a problem of global importance, such as malaria. This is a potentially fatal disease caused by parasites transmitted to humans by biting infected female mosquitoes of the genus Anopheles; control is mainly due to combating the vectors. However, extensive use of insecticides has subjected mosquitoes to intense selection pressure, resulting in the development of physiological and behavioural resistance; therefore, we propose to study the dynamics of resistant and non-resistant infected mosquitoes. Our contribution is a system of ordinary differential equations in which non-resistant mosquitoes are assumed to have a logistic growth and exit upon developing resistance. In addition, we assume that resistant mosquitoes can have non-resistant offspring and only natural death; we performed a stability analysis of the model, allowing us to conclude that both types of mosquitoes always exist beyond a certain threshold. In addition, we performed a numerical study considering different levels of insecticide represented by the deaths it produces.