Partial differential equations is a huge part of applied mathematics and it has a long history which connects to several other topics within mathematics. One of the most popular methods of solving these equations is known as the finite element method, which is a numerical method used to approximate a solution to a given PDE. In this thesis we will investigate how the theory behind weak solutions for partial differential equations connect with this solution method. It will also become apparent that there is a lot of deep underlaying theory to be covered to sucessfully make such an investigation.