Let the elliptic problem of form $-\Delta u + u = f$ on $\Omega\subset\mathbb{R}^2$, where $\Omega$ is a sufficiently smooth bounded domain. Let $u = 0$ at $\partial \Omega.$ The finite element method for that problem reads as follows : Find $u_h\in S_h$ such that
\begin{equation}
(\nabla u_h, \nabla v_h) + (u_h, v_h) = (f, v_h),\;\;\forall v_h\in S_h,
\end{equation}
where $S_h$ finite dimensional subspace of $H^1_0(\Omega).$ If we define the discrete Laplacian $-\Delta_h\,:\,S_h\to S_h$ such that $(-\Delta\psi, \phi) = (\nabla \psi, \nabla \phi),\,\,\forall \psi,\phi\in S_h$ and the orhogonal projection $P_h\,:\,L_2(\Omega)\to S_h,$ we can write the above system in operator form in $S_h.$, i.e.,
\begin{equation}
(-\Delta_h + I)u_h = P_hf.
\end{equation}
Now, if we have pure Neumann boundary conditions, i.e., $\nabla u \cdot n = g$ in $\partial \Omega.$ It is well-known that the finite element formulation of this problem reads as follows
Find $u_h\in S_h$ such that
\begin{equation}
(\nabla u_h, \nabla v_h) + (u_h, v_h) = (f, v_h) + (g, v_h)_{\partial \Omega},\;\;\forall v_h\in S_h,
\end{equation}
where $S_h$ finite dimensional subspace of $H^1(\Omega).$
My question:
How can I write the latter problem in operator form? Should I define a special projection for function $g$?
Best Answer
In your notation, if we also defined $P_{h,\partial\Omega}$ the $L^2$ projection on $\partial\Omega$, then we could write the whole thing as $$(-\Delta_h+I)u_h = P_hf+P_{h,\partial\Omega}g.$$