Solving PDEs via FEM: A Gut Check of the Procedure

[Vanilla Gorilla]

TL;DR Summary

I want to understand the process computers goes about solving PDES in FEM, and/or how people solve PDES in FEM without the aid of software.
I basically just want to check if my understanding of the method - described in the Google Doc - is correct, before I use software.

Hi, I am a newcomer to FEA/FEM. I am aware that for any practical purpose, software is used to solve problems. However, before I utilize software, I want to understand the process the computer goes about doing. This document is simply an attempt to summarize the solving procedure in a concise manner, according to my understanding. Because I am new, this understanding should have flaws, and I would just like someone to check it over to make sure everything looks right, and/or to clear up some of the confusions I have, which are specified in the document.

https://docs.google.com/document/d/...uKhTJV43hpqS4B30Q/edit#heading=h.d4e1rnx82ic6

I know it's 12 pages, but it's not actually very long to read, most of the pages are taken up by pictures.Any help is much appreciated!
P.S., I'm not always great at articulating my thoughts, so my apologies if this question isn't clear. Also, I know this isn't high school material, but I am currently in high school, which is why I made my level "Basic/high school level."

Note to moderators: I think that is the most appropriate forum for this post. However, if not, apologies, and please feel free to move at your own discretion. :)