The finite element method is difficult to understand when studying all of its concepts at once. Therefore, I explain the finite element method in three levels with increasing complexity. At each level, different shape functions are introduced for discretizing the solution function of the given differential equation. At the first level, the shape functions are defined globally. At the second level, the shape functions are defined at the element level. And at the third level, the shape functions are defined on a reference element. Along the way, the concepts of assembly, integration by substitution and Gaussian quadrature are explained.
I am using the word ansatz for the assumed form of the solution function u. Although ansatz is a German word, it is also frequently used in English, especially in the mathematical context. https://en.m.wikipedia.org/wiki/Ansatz
Video about the weak formulation: https://youtu.be/xZpESocdvn4
This is my contribution to the Summer of Math Exposition (SoME) 2024. #SoMEpi
0:00 Introduction
2:45 Level 1
19:37 Level 2
26:33 Level 3
38:21 Summary
Keywords: finite element method, finite element analysis, partial differential equation, Poission problem, weak formulation, discretization, assembly, isoparametric concept, reference element, Gaussian quadrature
Music: Swans In Flight - Asher Fulero
