What I found amazing about FEM is not the detail of how to implement it in code, but all the PDE theory & approximation theory -- how you can express the original continuous-domain infinite dimensional problem in a weak form using an infinite dimensional space of test functions, then approximating the weak form of the original problem with a finite dimensional Galerkin approximation, using a finite dimensional space of test functions, and use that to define a finite dimensional system of equations to solve. Then the theory for under what conditions you can guarantee that approximate solutions obtained from your finite dimensional approximation converge toward the true solution, as you increase the mesh resolution, and how fast the convergence rate will be.
Some of this is summarised in this paper in 2. model problem & 3. Galerkin discretisation of the problem, but not in a way that will communicate the mathematical ideas to anyone who hasn't already taken a course on the theory -- probably need a couple of courses on real analysis & a course on PDE as pre-reqs.
Yup. I'm trying to learn computational fluid dynamics for fun, and realised it's basically all numerical methods for PDEs, with some fluid dynamics-specific shortcuts. I know I've previously seen this stuff in relation to pricing derivatives in finance.
It's clear the underlying techniques are very powerful any time you have a thing whose rate of change varies as other things change. Once I understand everything better I will try it on e.g. capacity planning cloud resources and such.
Some of this is summarised in this paper in 2. model problem & 3. Galerkin discretisation of the problem, but not in a way that will communicate the mathematical ideas to anyone who hasn't already taken a course on the theory -- probably need a couple of courses on real analysis & a course on PDE as pre-reqs.