Solving PDEs via FEM: A Gut Check of the Procedure

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Vanilla Gorilla
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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. :)
 

1. What is FEM and how does it relate to solving PDEs?

FEM stands for Finite Element Method, which is a numerical technique used to solve partial differential equations (PDEs) by dividing the domain into smaller elements and approximating the solutions within each element. FEM is a powerful tool for solving complex PDEs that do not have analytical solutions.

2. How does the "Gut Check" procedure help in solving PDEs via FEM?

The "Gut Check" procedure involves systematically verifying the accuracy of the finite element solution by comparing it to known analytical solutions or benchmark problems. This helps in validating the numerical implementation and ensuring that the results are reliable and accurate.

3. What are some common challenges faced when using FEM to solve PDEs?

Some common challenges include choosing appropriate element types and mesh sizes, dealing with numerical instabilities, and ensuring convergence of the solution. Additionally, accurately modeling boundary conditions and material properties can also be challenging.

4. How can one determine the accuracy of the finite element solution?

One way to determine the accuracy of the finite element solution is to compare it to analytical solutions or benchmark problems, as well as performing convergence studies by refining the mesh and observing the changes in the solution. Additionally, error estimation techniques such as Richardson extrapolation can be used to quantify the error in the solution.

5. What are some best practices for using FEM to solve PDEs?

Some best practices include carefully selecting appropriate element types and mesh sizes, verifying the implementation through validation tests, and performing sensitivity analyses to understand the impact of different parameters on the solution. It is also important to document the modeling assumptions and numerical settings to ensure reproducibility of the results.

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