Finite Element Analysis is a mathematical tool very extended among engineers. However, after more than a year researching on the topic of computer simulation, where FEA plays such an important role, I haven't yet found a satisfactory explanation on how they really really work...
Hopefully by means of this post I get to clarify some of the topics, trying to remove excessive algebraic and mathematical verbosity that is so annoyingly present everywhere.
The main background of FEM is that of structural engineering in the 60s. Engineers are very practical people, so initially they devised a system which allowed them to set algebraic equations where the relations between different points or "nodes" of their structures could be set.
These algebraic relations were further proven to be of many different types, not only structural, so also thermal relations could be formulated and many others: all one needed to analyze was a proper discretizacion of the space in the form of a mesh with nodes to relate to each other.
http://stochasticandlagrangian.blogspot.com/2011/07/what-does-shape-function-mean-in-finite.html
https://www.ccg.msm.cam.ac.uk/images/FEMOR_Lecture_2.pdf
https://www.geophysik.uni-muenchen.de/~igel/Lectures/NMG/08_finite_elements_basisfunctions.pdf