Let $X$ be a smooth complex variety (not necessarily compact) and let $D$ be a normal crossings divisors with components $D\_1$, $D\_2$, ..., $D\_N$. For a set of indices $I$, let $D\_I = \bigcap\_{i \in I} D\_i$ and let $D^{\circ}\_I = D\_I \setminus \bigcup\_{J \supsetneq I} D\_J$. I would like to compute $H^{\ast}(X)$ in the case that I know all of the $H^{\ast}(D\_I^{\circ})$.


It seems to me that there should be a spectral sequence whose first page is $\bigoplus\_{\#(I)=p} H^{q-p}(D^{\circ}\_I)$. If $I' \supset I$ with $\#(I') = \#(I)+1=p+1$, then the map $H^{q-p}(D^{\circ}\_{I}) \to H^{q-(p+1)}(D^{\circ}\_{I'})$ would given by the Gysin map (up to standard sign issues).


Does this sequence have a name, or is it a special case of something which has a name? Where can I read about it?



A spectral sequence for computing cohomology of a space from that of its strata
Let $X = T\_n \supset T\_{n-1} \supset \cdots \supset T\_{-1} = \varnothing$ be a topological space filtered by closed subspaces, where for simplicity I assumed the filtration bounded. Then there is a spectral sequence
$$ E\_1^{pq} = H^{p+q}\_c(T\_p \setminus T\_{p-1}) \implies H^{p+q}\_c(X).$$
(Some mild point-set assumptions are required for this and what follows to be true, but let me disregard that.) As you noted in a comment, this gives the spectral sequence you want by Poincaré duality.


Usually, the spectral sequence of a filtered space is written in terms of ordinary cohomology rather than cohomology with compact support. One reason is that the compactly suppported spectral sequence is a special case of the usual one. Recall that if $U \subset \overline U$ is any compactification of a space $U$, then $H^\bullet\_c(U) = H^\bullet(\overline U, \partial U)$, where $\partial U = \overline U \setminus U$. So if we choose an arbitrary compactification $X \subset \overline X$ and let $\overline T\_p$ be the closure of $T\_p$ in $\overline X$, then we get a filtration 
$$\overline X = \overline T\_n \cup \partial X \supset \overline T\_{n-1} \cup \partial X \supset \overline T\_{n-2} \cup \partial X \supset \cdots \supset \overline T\_{-1} \cup \partial X = \partial X.$$


Now the usual spectral sequence of a filtration reads
$$ E\_1^{pq} = H^{p+q}(\overline T\_p \cup \partial X,\overline T\_{p-1}\cup \partial X) = H^{p+q}(\overline T\_p,\overline T\_{p-1}) \implies H^{p+q}(\overline X, \partial X),$$
and this is the spectral sequence we wanted. 


(Often one chooses $\overline X$ and $\overline T\_p$ to be the one-point compactification of $X$ resp. $T\_p$, and then the above is just the spectral sequence for the reduced cohomology a filtered *based* space. But allowing yourself to use any compactification is useful in the algebraic setting, which you are interested in.)


Here's an alternative "sheafy" derivation of the spectral sequence. The constant sheaf $\mathbf Z\_X$ is filtered:
$$\mathbf Z\_X = \mathbf Z\_{T\_n} \supset \mathbf Z\_{T\_{n-1}} \supset \cdots \supset \mathbf Z\_{T\_{-1}} = 0$$ where I denote by $\mathbf Z\_A$ the pushforward of the constant sheaf on the subspace $A$. The successive quotients in the associated graded for this filtration are of the form ${j\_n}\_!\mathbf Z$, where $j\_n$ is the locally closed inclusion of $T\_n \setminus T\_{n-1}$. Taking compactly supported cohomology of this filtered object gives rise to a spectral sequence which is exactly the one we want. A useful reference for making sense of this is the section about filtered objects and spectral sequences in Lurie's "Stable $\infty$-categories" (Chapter 1 of Higher Algebra). 


To answer a question you made in the comments, the spectral sequence of a filtered space is indeed compatible with mixed Hodge structure, when $X$ is an algebraic variety filtered by closed subvarieties. So the one you consider (by Poincaré duality) is compatible up to Tate twist. I don't know what is the canonical reference for this fact, but Sections 3 and 4 of Arapura's "The Leray spectral sequence is motivic" proves rather generally that the spectral sequence of a filtered algebraic variety is compatible with all kinds of extra "motivic" structure.


