  Let X be a non-compact Calabi-Yau manifold and f be a holomorphic function on
X with compact critical locus. We introduce the notion of f-twisted Sobolev
spaces for the pair (X,f) and prove the corresponding Hodge-to-de Rham
degeneration property via L2-Hodge theoretical methods when f satisfies an
asymptotic condition of strongly ellipticity. This leads to a Frobenius
manifold via the Barannikov-Kontsevich construction, unifying the
Landau-Ginzburg and Calabi-Yau geometry. Our construction can be viewed as a
generalization of K.Saito's higher residue and primitive form theory for
isolated singularities. As an application, we construct Frobenius manifolds for
orbifold Landau-Ginzburg B-models which admit crepant resolutions.
