Modélisation et analyse numérique de problèmes de réaction-diffusion provenant de la solidification d'alliages binaires

In this work, we are interested in elaborating models and numerical methods in order to study some phenomena which arise in the solidification of binary alloys. We propose two distinct models based on the mass and energy conservation laws as well as on classical thermodynamics. The first model is based on the irreversible process theory and therefore requires the computation of the entropy of the system to complete the description. To obtain this quantity, we develop a formalism and a method which yields numerical results. This construction is made so that the entropy function inherits some interesting mathematical properties from physical requirements. These properties allow us to elaborate and analyze mathematically an original scheme using a time discretization depending on a real parameter to solve the solidification problem. In particular, we are able to prove that the scheme is stable for all values of the time step if the parameter is chosen correctly. Furthermore, we give some numerical results to support the theory. The second model, called phase-field model, is used to describe dendrites' formation during the solidification of binary alloys. The nature of the problem constrains us to use very fine meshes in some physical regions. To reduce the number of discrete unknowns, we develop an adaptive mesh strategy based on an ad-hoc error estimator. We make numerical tests showing that the method converges on regular meshes and that the estimator is in good agreement with the true error. We also show that the mesh refinement strategy gives good results in academic and physical cases.