Análise numérica da equação de Allen-Cahn

Abstract

In the following thesis, we will study the Allen-Cahn equation ut = Δu − ϵ−2f(u). We will begin by showing the well-posedness of the boundary value problem induced by this equation with Neumann boundary conditions. In addition, we will prove that its solution satisfies a Maximum Principle and some regularity proprieties. Next, we will move to the numerical analysis and study a semi-implicit discretization in time, showing that it is unconditionally stable and that its error grows only polinomially in ϵ−1 if the time-step satisfies some hypothesis. Finally, we will introduce a spatial discretization, using the finite element method, and show that, with certain adaptations in the arguments, it is also possible to obtain an error estimative that depends only polinomially in ϵ−1.