For a variety $X/k$ the standard way of defining de Rham cohomology $H^i_{\mathrm{dR}}(X)$ is as the hypercohomology $\mathbb{H}^i(\Omega^{\bullet}_{X/k})$ of the de Rham complex, and this requires $X$ to be smooth. Hartshorne has extended the definition of $H^i_{\mathrm{dR}}$ to singular varieties, but using techniques that are too complicated for me. So I would appreciate if someone could help me to find out if the two definitions agree in certain circumstances.
Does the naive definition agree with actual de Rham cohomology:
$1)$ when $X$ has normal crossing singularities?
$2)$ when $X$ is a curve with normal crossing singularities?
$3)$ when $X$ is a union of $\mathbb{P}^1$ with transverse intersection?
If these are not true, do we at least have agreement of the Hodge filtration $H^p(\Omega^q_{X/k})$? I'm happy to restrict to projective varieties.
Thanks.