Mathematical Models and Methods in Applied Sciences

We analyze approximation properties of dimension reduction models that are based on mixed principles. The problems of interest are elliptic PDEs in thin domains. The goal is to obtain estimates that take into account both the thickness of the domain and the order of the model. The techniques involved do not require the models to be energy minimizers, and are based on asymptotic expansions for the exact and model solutions. We obtain estimates in several norms and semi-norms, and also interior estimates (which disregards boundary layers).

Slender beams with small cracks described by Γ limits: a description of an elastic-perfectly plastic beam or rod is obtained as a variational limit of 2D or 3D bodies with damage at small scale satisfying the Kirchhoff kinematic restriction on the deformations.

We consider the three-dimensional scalar problem of acoustic propagation in a muffler. We develop and analyze a Fredholm-type formulation for a stationary fluid in the time-harmonic setting. We prove a homogenization result for a muffler containing periodically perforated ducts. Essentially, the perforated boundaries become completely transparent when the period of perforations, which is assumed to be of the same order as the size of perforations, tends to zero. We also derive a homogenized impedance condition when the perforated duct is coated by an absorbing material. We present numerical examples in two dimensions, obtained from coupling finite elements in the muffler with modal decompositions in the inlet and outlet ducts, which demonstrate the limiting validity of the theoretical results.

A selection mutation equations model for the distribution of individuals with respect to the age at maturity is considered. In this model we assume that a mutation, perhaps very small, occurs in every reproduction where the noncompactness of the domain of the structuring variable and the two-dimensionality of the environment are the main features. Existence of stationary solutions is proved using the theory of positive semigroups and the infinite-dimensional version in Banach lattices of the Perron Frobenius theorem. The behavior of these stationary solutions when the mutation is small is studied.

In this work we introduce a new a posteriori error estimate of hierarchical type for the advection-diffusion-reaction equation. We prove the equivalence between the energy norm of the error and our error estimate using an auxiliary linear problem for the residual and an easy way to prove inf–sup condition.