Let $(X,A)$ be a pair of manifolds, where $A \subseteq X$. Then we can define the relative cohomology of the pair $H_{\bullet}(X,A)$ to be the homology of the chain complex $C_{\bullet}(X)/C_{\bullet}(A)$, where $C_{\bullet}(A), C_{\bullet}(X)$ are the singular chain complexes associated to $A$ and $X$ respectively.
We can also define relative de Rham cohomology of the pair $(X,A)$. I believe that for this you consider the double complex $\Omega^{\bullet}(X) \to \Omega^{\bullet}(A)$ (where we are pulling back forms by the inclusion map), take its total complex and then take its cohomology: $H^{\bullet}_{dR}(X,A)$.
From what I understand these two theories are not dual. So my question is: which cohomology theory is dual to the relative homology of the pair $(X,A)$.