Can we approximate a square integrable function with square integrable Laplacian in this way by smooth functions?

Question

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set with a smooth boundary. Suppose that $f\in L^2(\Omega)$ satisfies $\Delta f\in L^2(\Omega)$ (in the sense of distributions/weak derivatives), where $\Delta=\sum_{j=1}^n{\partial_j}^2$ is the Laplacian. Does there exist a sequence of compactly supported smooth functions $\{f_k\}_{k\in\mathbb{N}}\subset C^\infty_c(\mathbb{R}^n)$ defined on the whole of $\mathbb{R}^n$ such that the following two convergences hold: \begin{cases} f_k\to f&\text{in }L^2(\Omega)\\ \Delta f_k\rightharpoonup\Delta f&\text{in }L^2(\Omega) \end{cases}as $k\to\infty$? Note that the former convergence is in the strong $L^2(\Omega)$ topology while the latter is in the weak $L^2(\Omega)$ topology.