Let's consider the weak formulation of the following linear wave equation:
Let $\Omega \subset \mathbb{R}^d$ ($d \in \{2,3\}$) be some open, bounded Lipschitz domain. Denote $H^1_0(\Omega)$ as the Sobolev space with zero trace and $H^{-1}(\Omega)$ it's dual space.
Search a function $u \in L^2\left(0, T; H^1_0(\Omega)\right)$ with $u' \in L^2\left(0, T; L^2(\Omega)\right)$, $u'' \in L^2\left(0, T; H^{-1}(\Omega)\right)$ and
$$ \begin{equation} \langle u''(t), v \rangle_{H^{-1}(\Omega) \times H^1_0(\Omega)} + \langle \nabla u(t), \nabla v\rangle_{L^2(\Omega)} = \langle f(t), v\rangle_{L^2(\Omega)}, \end{equation} $$ for almost every $t \in (0,T)$ and all $v \in H^1_0(\Omega)$ and with initial conditions $$ \begin{equation} u(0) = u_0, \ \ u'(0)=v_0, \end{equation} $$ with data $f \in L^2\left(0, T; L^2(\Omega)\right)$, $u_0 \in H^1_0(\Omega)$ and $v_0 \in L^2(\Omega)$.
The corresponding Galerkin approximation of the above problem uses, for instance, $N$-dimensional subspaces $V_N$ of $H^1_0(\Omega) \subset L^2(\Omega)$ spanned by the eigenfunctions of the Laplace operator (or taking $V_N$ as Finite Element spaces of piecewise polynomials). It then reads:
Search a function $u_N \in L^2\left(0, T; V_N\right)$ with $u_N' \in L^2\left(0, T; V_N\right)$, $u_N'' \in L^2\left(0, T; V_N\right)$ and
$$ \begin{equation} \langle u_N''(t), v_N \rangle_{L^2(\Omega)} + \langle \nabla u_N(t), \nabla v_N\rangle_{L^2(\Omega)} = \langle f(t), v_N\rangle_{L^2(\Omega)}, \end{equation} $$ for almost every $t \in (0,T)$ and all $v_N \in V_N$ and with initial conditions $$ \begin{equation} u_N(0) = \Pi^{L^2(\Omega)}_{V_N}(u_0), \ \ u_N'(0)=\Pi^{L^2(\Omega)}_{V_N}(v_0), \end{equation} $$ where $\Pi^{L^2(\Omega)}_{V_N}$ denotes the $L^2$ orthogonal projection of the functions onto $V_N$. This is a system of linear ODEs with a unique solution in $H^2\left(0,T; V_N\right)$, i.e. $u_N \in C^1\left([0,T]; V_N \right)$ and $u_N'' \in L^2\left( 0,T; V_N \right)$.
From basic theory of hyperbolic equations ("Partial Differential Equations" by Evans or "Nonlinear Functional Analysis and Its Applications IIA" by Zeidler) one usually proves the existence of a weak solution $u$ by using the existence of $u_N$ and then arguing by weak convergence $$ \begin{eqnarray} u_N \rightharpoonup u \ \ &\textrm{weakly in}& \ L^2 \left( 0, T; H^1_0(\Omega)\right), \\ u_N' \rightharpoonup u' \ \ &\textrm{weakly in}& \ L^2 \left( 0, T; L^2(\Omega)\right). \end{eqnarray} $$ One can also show the continuity $u \in C\left( [0,T]; H^1_0(\Omega) \right)$ and $u' \in C\left( [0,T]; L^2(\Omega) \right)$.
Now my question is: Is it also possible to show stronger convergence results for these Galerkin approximations?
For example strong convergence $u_N \rightarrow u$ in $L^2 \left( 0, T; H^1_0(\Omega)\right)$ and $u_N' \rightarrow u'$ in $L^2 \left( 0, T; L^2(\Omega)\right)$ or even better, since $u, u_N \in C\left( [0,T]; H^1_0(\Omega) \right)$ and $u', u_N' \in C\left( [0,T]; L^2(\Omega) \right)$ is it also possible to deduce convergence in these space? That means
$$ \begin{equation} \max_{0 \leq t \leq T} \left( \| u(t) - u_N(t) \|_{H^1_0(\Omega)} + \| u'(t) - u_N'(t) \|_{L^2(\Omega)} \right) \rightarrow 0, \ \ \textrm{as $N \to \infty$} \end{equation}, $$
without any additional regularity assumptions.
I would be grateful for some arguments or literature that treats these convergence results.
Yes, this exists and even with rates. A very complete convergence proof is contained in Chapter 8 of the book by Raviart and Thomas.
Link on Google books