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Comment by mpoteat | Hacker News Reader

This made as little sense to me as it did when I was talking the Finite Element Methods class during graduate school.

Still don't quite understand why you can't just use Runge Kutta methods to numerically solve these problems. I became quite good at manipulating the symbols to derive variational solutions while having absolutely no idea what any of it meant.

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Runge-kutta works when you’re given an initial condition and the derivative is with respect to just one variable (so you’re given f(t) just at t=0). What do you do when you’re given a boundary condition and the derivative is with respect to many variables? This.

I mean, you can also just do finite differences. Of course Galerkin opens up the very “rich” design space of finite elements methods.

Finite differences are simple in cubes and domains that can be mapped to one. Not so much in an arbitrary tetrahedral mesh.

I think variational problems are more naturally understood through energy minimization. You start with the energy of the system and try to minimize it via derivatives. Then you arrive at a variational problem. The differential equation is then more of an afterthought.

Finite differences is the PDE analogue of Runge-Kutta, and it is certainly used (in CFD for example). However, finite element methods have several advantages:

* It can handle PDEs on domains with complicated geometries, while finite differences really prefer rectangular domains. This consideration doesn't apply to ODEs which are always solved on one-dimensional intervals.

* For any numerical approximation it is important to have convergence guarantees, and as the blog post mentions, the analysis is much more well understood for finite elements, particularly on irregular geometries. Strang and Fix's 1973 book is the classic reference here.

If your solution is troublesome in some way (discontinuous, singular), then you can't rely on methods that need it to have derivatives

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