Wedge product between relative de Rham cohomology and de Rham cohomology on a subspace

Question

Under appropriate conditions on $U\subset X$ (what are they?), is there a well-defined wedge product $$ \wedge: H^k(X,\overline U)\oplus H^l(X\setminus U)\to H^{k+l}(X,\overline U), $$ and if so, is there a name for it? What is the canonical way to define it?