If $\mathbf{f} \in H^{1/2}(\Omega)$ for a bounded domain $\Omega$ with smooth boundary, does the elliptic regularity for the Laplacian guarantee that the solution to the elliptic problem in divergence form $$ \begin{cases} \Delta \phi = \operatorname{div}(\mathbf{f}) & \mbox{ in } \Omega,\\ \phi = 0 & \mbox{ on } \partial \Omega, \end{cases} $$ satisfies $\phi \in H^{3/2}(\Omega)$ ?
Remark 1: The only literature I know is "Non-Homogeneous Boundary Value Problems and Applications I" by Lions & Magenes, e.g. pages 188-199 below Remark 7.2, but the setting is so involved that I am not sure if it exactly answers my question. Probably it boils down to the question whether $\operatorname{div}(f) \in \Xi^{-1/2}(\Omega)$ holds for the spaces $\Xi^{s}(\Omega)$ defined on Page 172 in the latter book, which I am not sure whether it is trivial since $H^{1/2}(\Omega) \subseteq \Xi^{1/2}(\Omega)$ is a dense inclusion according to that reference.
Remark 2: I believe for the Dirichlet problem it is not required, but it would be perfectly fine for me to assume (if necessary in a weak sense) that $\int_{\Gamma_l} \mathbf{f} \cdot \mathbf{n} = 0$ on each connected part $\Gamma_l \subseteq \partial \Omega$. It would even be fine for me to assume that $\mathbf{f}$ has a well-defined normal trace(which is not automatically guaranteed by the setting above)