Numerical simulation of fractional models for fractures and damage problems

Recently, several problems of fractures, damages and fatigue of materials have been modelled by fractional differentials operators. An accurate and efficient numerical solution of these models is a fundamental but difficult issue, since many standard methods exhibit a slow error decay. Here we illustrate one and two step spline collocation methods for fractional differential problems. These methods have a fast error decay rate, if a suitable graduated mesh is adopted. Moreover, they have strong stability properties, since it is possible to set the method parameters to avoid stepsize restrictions due to stability. Therefore, spline collocation methods can accurately simulate also stiff fractional differential problems, at a reasonable computational cost. In this talk we will talk about the main theoretical results regarding convergence and linear stability, and we will show a number of significant numerical experiments.