In every textbook and at school, one can see the following way to solve for a Poisson equation using FEM:
- (1) start with $\Delta u = b$
- (2) obtain the weak formulation : $\int \Delta u~v~dx = \int b~v~dx$
- (3) integrate by parts to get : $-\int \nabla u \nabla v = \int b~v~dx$
then decompose $u$ and $v$ on finite element basis to get the linear system to solve.
My question is about point (3) : why is it necessary ? Why can't you directly use the $\Delta$ as it is, as it could be any other linear operator (otherwise, how do you solve when you have other linear operators ?).
The motivation for this integration by part is never mentionned (including wikipedia etc.).
Edit: to be more precise, keeping the $\Delta$ still allows to write the problem as $A(u,v)=L(v)$ with $A$ a bilinear form and L a linear form... why do we need to convert it to something else?