Lecture Notes: The Galerkin Method
  These lecture notes introduce the Galerkin method to approximate solutions to partial differential and integral equations. We begin with some analysis background to introduce this method in a Hilbert Space setting, and subsequently illustrate some computational examples with the help of a sample matlab code. Finally, we use the Galerkin method to prove the existence of solutions of a nonlinear boundary value problem. 
Raghavendra Venkatraman
math.AP math.FA math.NA