Let $\pi:Y\rightarrow X$ be a $C^\infty$ fibered manifold (all constructions, unless otherwise stated are over the smooth manifold category) with $\Omega_k$ the sheaf of (smooth) $k$-forms on $Y$.
This sheaf has a natural filtration, namely let $\mathscr H$ denote the ideal sheaf generated algebraically by the $\pi$-horizontal $1$-forms and set $\mathscr H^r_k$ to be the sheaf of all $k$-forms that belong to the $r$th power of $\mathscr H$. Then $$ \Omega_k=\mathscr H^0_k\supseteq\mathscr H^1_k\supseteq\dots\supseteq\mathscr H^k_k\supseteq 0 $$ is a descending filtration compatible with $d$, so there is in particular a spectral sequence associated to this filtration. The $E_0$-sheet consists of the quotient sheaves $$ \mathscr V_{k,l}:=\mathscr H_{k+l}^{l}/\mathscr H_{k+l}^{l+1} $$with differentials induced from $d$ that go as $$ d:\mathscr V_{k,l}\rightarrow\mathscr V_{k+1,l}. $$
The cohomology of this sequence can be computed by the de Rham-Weil theorem since it is easy to see that the sheaves $\mathscr V_{k,l}$ are soft and the complexes $(\mathscr V_{\ast,l},d)$ are exact everywhere except at $\mathscr V_{0,l}$. Let $\mathcal S_l$ denote the inverse image sheaf $\pi^\ast\Omega_l^X$, where $\Omega^X_l$ is the sheaf of $l$-forms on $X$.
Then $$ 0\rightarrow\mathcal S_l\rightarrow\mathscr V_{0,l}\rightarrow\mathscr V_{1,l}\rightarrow \dots $$ is a soft resolution of the sheaf $\mathcal S_l$, hence $$ H^k(\mathscr V_{\ast,l}(Y),d)=H^k(Y,\mathcal S_l), $$i.e. the cohomology of the $\mathscr V_{\ast,l}$-sequence is isomorphic to the cohomology of the sheaf $\mathcal S_l$.
My question is essentially how does the spectral sequence associated to the above filtration relate to the Serre spectral sequence? Is it the same (at least when $\pi:Y\rightarrow X$ is a fibre bundle)?
I am much more familiar with differential geometry than algebraic topology, so I don't know a lot about the Serre spectral sequence aside from the basic idea and I have not seen the construction I have outlined above elaborated anywhere.
In particular, can we say any more about the cohomology spaces $H^k(Y,\mathcal S_l)$ in particular how they relate to the cohomologies $H^k_{\mathrm{dR}}(Y)$, $H^k_{\mathrm{dR}}(X)$ and $H^k_{\mathrm{dR}}(F)$ (when $\pi:Y\rightarrow X$ is locally trivial with typical fibre $F$)?
If this is not the Serre spectral sequence in disguise then does the construction I gave have a standard name?
