Thank you for your document, and sorry for the delay (my brain runs so slow...).
From the first part of your demonstration I see that from the poisson equation integrated by parts, and after applying the Gauss theorem, the boundary conditions appears in the main equation. For clarity, I think that it would be better to use a different notation for the test function because 's' is also used in 'ds' as the surface element, unless I missed something again...
I am confused that in , at the beginning of chapter 3, equation (3.16) which is more general (i.e. not only for the poisson problem) has the main equation and boundary conditions just added together with the justification that if it works for all v (the test functions) it is equivalent to the two equations (is that because the two terms are not integrated over the same domain : the volume and the surface of the volume ?)... integration by parts is suggested only after.
So in that case, having the boundary conditions inside the weak form is independent of the type of the differential equation, and adding the main equation and the boundary conditions is just a way to solve the two equations at once...
Can you help ?
 The Finite Element Method - Fifth Edition - Volume 1 : The Basis - ZIENKIEWICZ & TAYLOR - ISBN 0 7506 5049 4
Le vendredi 31 août 2012 20:59:53 UTC+2, Alec Kalinin a écrit :
In the previous topic  I promised to David to write the description of the weak form terms for the Poisson equation. So, here is it. You can freely use and modify this description without any restrictions. But I am not very sure that the text is very accurate.