Galerkin methods for parabolic problems
The Galerkin finite element method of lines is one of the most popular and powerful numerical techniques for solving transient partial differential equations of parabolic type. The Galerkin finite element method of lines can be viewed as a separation-of-variables technique combined with a weak finite element formulation to discretize the problem in space. This leads to a stiff system of ordinary differential equations that can be integrated by available off-the-shelf implicit ordinary differential equations (ode) solvers based on implicit time-stepping schemes such as backward differentiation formulas (BDF) or implicit Runge-Kutta (IRK) methods.
Adjerid, Slimane and Baccouch, Mahboub, "Galerkin methods for parabolic problems" (2010). Mathematics Faculty Publications. 6.