**An alternative.** 

The given problem is equivalent to the minimization problem
 \begin{align}
 \mathcal{J}(v_k)=  \min_{v\in \mathcal{V}_k}  \mathcal{J}(v)\quad \text{with}\quad  \mathcal{J}(v):=  \frac12 \int_\Omega|\nabla v|^2 dx + \int_\Omega G(v)d x + \int_\Omega fvd x
 \end{align}
and we define the function $G(v)=  \int_0^v\zeta(\tau)\tau d \tau $.  Note that $G$ is non-negative since $\zeta(\tau)\geq 0$ and that $\mathcal J$ is continuous on $\mathcal V_k$. Using the Poincar\'e-Friedrichs inequality we find that $\mathcal{J}(v)\to \infty$, as $\|v\|_{L^2(\Omega)} \to \infty$ and $v\in \mathcal{V}_k$. Which implies the existence of a minimizer $v_k\in \mathcal{V}_k$ of \mathcal{J}, since we are in finite dimension space.