I am learning finite element method and I would like to compute the weak formulation of the following problem:
$-\nabla^2u + u = f$
with the boundary conditions $u=u_0$ (Dirichlet) and $\nabla u \cdot n=g$. I am able to write the weak formulation of the Poisson equation and I did this:
$-\int_\Omega (\nabla^2 u)v \,d \Omega +\int_\Omega uv\,d \Omega = \int_\Omega fv\,d \Omega$.
Is this correct? I know that the first term in the LHS can be further "simplified" as in the Poisson equation, but I want to be sure that the second term in the LHS is correct. Thank you.