Let $\Omega=(0,1)^2$ and consider the problem : find $u:\overline{\Omega} \to \mathbb{R}$ such that
$\begin{align} -\Delta u + u_{x}+3u = 1~in~\Omega,\\ u = g ~on~\partial\Omega \end{align}$
where function $g$ is given. Consider the triangulation $T_h$ of $\Omega$ consisting a uniform triangles with diameter $h$. Let $X_h=\{v \in C(\overline{\Omega}): v|_T \in P_k(T) ~\forall T\in T_h\}$, where $k\in\mathbb{N}$.
I woul like to derive the best error estimate $||u-u_h||_{1,\Omega}$, where $u_h \in X_h$. Fisrt of all, I derive weak formulation:
I seek $u\in H^1(\Omega)~$:
$\begin{align} a(u,v)=<f,v> \forall v \in H^1_0(\Omega)\\ u-h\in H^1_0(\Omega), \end{align}$
where
$a(u,v)=\int_\Omega \nabla u \nabla v + u_x v + 3uv ~dx$,
$<f,v>=\int_\Omega v dx$,
$h\in H^1(\Omega)$ such that $h|_{\partial\Omega}=g$.
Now I have problem how to derive th best estimate of error in $H^1$-norm. Any help ?