In engineering sciences, we usually end up using either the second- or the fourth-order differential equations, and the MWR (the method of weighted residuals) works pretty well for them.
The question is: how about the first- and the third-order differential equations? Why don't we see any applications of MWR for these odd-ordered differential equations? What gives?
It has been quite some time (more than 1.5 years) that I had touched upon the topic of the physical bases of FEM in general, and of the general weighted residual (WR) approach in particular, at iMechanica (see here).
The position I then took was that there is no known physical basis at all for the WR approach---despite its loving portrayals in mathematical terms, or its popularity.
Further, I had also expressed (here and elsewhere) that a basis in physical principles existed for FEM only in a rather limited sense: wherever the energy interpretation was available for the model. (Note, this too is already at variance with what some of the authors have written in books.)