The Galerkin Finite Element Method is most simply described as the

The Galerkin Finite Element Method is most simply described as the Ritz method on
subdivided domains. The steps required for Galerking FEM are as follows.
1) Put the problem into variational formulation.
This step is the same as the Ritz method, where you multiply the equation that is in
differential form, by a test function. This is then integrated by parts to finish the
conversion of the differential form into variational form. Boundary conditions can
simplify these expressions if they are Dirichlet or homogenous Neumann conditions.
2) Create a finite element mesh and basis functions.
The finite element mesh is created by subdividing the domain into continuous elements
representing the entire domain, with nodes separating the elements. The basis functions
are weighing functions that are essentially interpolate between elements; when applied to
each element we get local basis functions. The local basis functions are then converted
into a global basis, by essentially creating relations to each node. This was summarized in
the video as local basis using elements, and the global having the element relations
converted into nodes.
3) The Galerkin Method
This portion states that an approximation of the desired solution is equal to the
summation of the values of the unknown at the nodes multiplied by the basis function at
each node. Additionally, an actual value of the test function is made to be a linear
combination of the identity combination of basis function. These create a system of
equations that can be used to solve for each distinct unknown value at each node. The
fact that the global basis functions are 0 at many points on the domain, simplifies the
system of equations greatly.
4) Solve the system of equations
This portion is self explanatory. The system of equations is solved to determine the
unknown value at each node. These are put back into the summation detailed in the
Galerkin method to obtain the approximate solution.
5) Verification
As this method obtains an approximate solution, care must be taken to ensure that the
results obtained are accurate.