Is $C^\infty_{0}(\bar{\Omega})$ dense in the hilbert space $W^{2,2}(\Omega)\cap W^{1,2}_{0}(\Omega)$

Question

Assume $\Omega$ is open bounded domain in $\mathbb R^n$

Is $C^\infty_{0}(\bar{\Omega})$ dense in the hilbert space $W^{2,2}(\Omega)\cap W^{1,2}_{0}(\Omega)$ with inner product $$(u,v)=\int_{\Omega} \Delta u \Delta v $$

I ask this question because I have proved an equality for $C^\infty_{0}(\bar{\Omega})$ functions and I want to prove it for the hilbert space mentioned with density argument