Fractal sets are “too rough" to define the classical Laplace operator as is the case for open subsets of R n . We can still however define the Laplace operator on fractals through various approaches, such as a probabilistic one studied by and an analytic one, via the use of Dirichlet forms, originating from the work of Kigami .In this thesis we will focus on Kigami’s analytic approach. We will use the slightly different convention of, that is the same in spirit to that of Kigami, but is perhaps more suited for considering things from the graph theoretic point of view. On the graph approximation Gn, we set Vn = ∪ i FiVn−1 and V∗ = ∪ iVi. Now, for functions u, v : Vn → R we can define the energy form