We consider the classical 1D Ising model, where the coupling constants and the external magnetic field take one of two possible values at each site, according to a substitution rule. We shall introduce (briefly) the idea of a trace map corresponding to the given substitution rule and how its dynamical properties can be used to investigate the partition function and, consequently, the free energy function of the given Ising model.
Adaptive finite element methods have been used for the solution of linear and non-linear elliptic partial differential equations since the 70s. However, only recently have rigorous convergence and (even stronger) contraction results been developed for a large class of interesting problems. In this talk, the basic adaptive finite element framework will be presented along with an overview of some convergence results. Also, a new efficient, reliable, and robust error estimator for problems in three dimensions will be presented along with numerical computations supporting its use.