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Galerkin Method - Why does integration-by-parts eliminate need to enforce Neumann boundaries?

I've posted this to MathStackexchange but I figured I'd also post here as well as I have yet to receive an answer on my original post, and that I would be more likely to encounter users of the Galerkin … From what I know of the Galerkin method, it is about approximating $u$, to a invertible linear system $Au=f$, as a vector in some subspace of the original square-integrable function space that $u$ resides …
Timothy Leong's user avatar