This chapter derives the finite element approach proposed by Arioli and Benzi (IMA J Numer Anal 38(3):1119–1163, 2018) and its application to the semidiscretization of parabolic equations. The discretization of metric graphs leading to the concept of extended graphs is reviewed to allow a structured representation of the finite element semidiscretization. The main new aspect elaborated for this book is the solution of the semidiscretized systems with implicit-explicit time stepping methods combined with a multigrid ansatz for the solution of the arising linear systems of equations. Moreover, an \(L^2(\Gamma )\) -error estimate will be derived which allows to use standard theory of Galerkin finite element methods for parabolic problems (see Thomée, Galerkin finite element methods for parabolic problems. Springer, 1997) to deduce estimates for the error between the solutions of the semidiscrete and the continuous problem.