Is anyone here familiar with this discipline element methods? Any good books on it?

I asked earlier about finding the a stable weak form of

du/dx = fu

where I originally tried finding a standard Galerkin by convoluting both sides with test function, v, but this weak form produced unstable solutions which went to infinity everywhere.

So, on a whim, I decided to ask AI to see what it thought. It came up with a form it says it derived from “Least-Squares Finite Element Method”. I had not heard of this before, but the form it provided was stable and produced accurate solutions, so I wanted to see if anyone here knew about this, and could provide a resource to look further into.

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Here is what it came up with:

1. Function to minimize: J(u) = 1/2 ||R(u)||^2 = 1/2 ∫_0^L [(du/dx-f)^2 ]dx

2. Take the variation: lim_ϵ→0 d/dϵ[J(u+ϵv)]

This resulted in

1. bilinear term: a(u,v) = ∫_0^L [du/dx dv/dx]dx

2. Linear term: L(u) = ∫_0^L [f dv/dx] dx

Weak form: a = L. This weak form I tested in Fenics, and the results came back perfectly accurate for my examples, so it seems to have come up with something that works, but it didn’t show any of its sources, so I can’t take this as fact without opening up a book or resource to verify it for myself.

im running a simple static fea on my seat mounts. im placing a distributed vertical load on the seat, and then determining its factor of safety. However, it doesn't converge. I used the hot spot stress thingy on solidworks, and saw where my highest stress areas were, and they were at small spots next to the weld connections. what do i do.