I'm trying to figure out how the second term appears here:
$$\int_\Omega \mathcal{L}(u) w \;d\Omega = \int_\Omega u \mathcal{L}^*(w) \; d\Omega + \int_\Gamma \left[S^*(w) G(u) - G^*(w)S(u)\right] d\Gamma$$
This is equation 1.5 from the textbook Boundary Element Techniques by Brebbia, I believe you can obtain it by transposing the inner product of the left hand-side but I don't understand how the second term appears.
The authors mention $S$ and $G$ are differential operators due to the integration by parts and are related to essential and natural boundary conditions. I know they can be obtained by doing integration by parts twice on the left-handside and they do provide an example but I fail to see why this is true in general