I was wondering if anyone could explain to me or send any relevant resources to help me understand the variational formulation formula in the FreeFEM magneto static example

In particular, I am interested in the 3D version of the formula. I am fairly new to variational formulation and FreeFE. I can’t seem to find any resources that derive or discuss a similar formula to that found in the example.

Sure. I am personally fond of the FEM book by Larson and Bengzon [0]. The derivation of the weak form is quite similar to the one presented for “The Time Harmonic Electric Wave Equation” (13.5 till 13.5.3) and there is also a section about “The Magnetostatic Potential Equation”.

The interesting thing about magnetostatic vector potential formulations is how to handle the nonuniqueness of the pontential A. If you compare the variational form displayed above the example freefem code with the form actually used in the code (in the 3D case), you should notice an additional div-div term. This regularization term is used in order to ensure a unique solution. There is a multitude of techniques to do this. A nice overview “About the gauge conditions arising in Finite Element magnetostatic problems” is given by Creuse et al. [1]. They also list some references for the div-div term.

I guess in general its the same recipe- take the differential form, multiple by the test
function, integrate by parts or use Green’s Theorem, and end up with an integral equation.
I am completely new to a lot of this, never done much FEM beyond playing with
some codes like libmesh, and I found if you are
interested in current topics google or google scholar on “weak form” and your favorite
equation turns up a lot. II can’t remember the exact search terms of where I put the URL
but I have found various specific problems- in one case some issue with a hybrid
approach to deal with Eddy currents in laminated cores ( I didn’t even know people
used these for anything any more lol ).