  In this paper, we establish the convergence for Gromov-Witten invariant of
elliptic orbifold $\mathbb{P}^1$ with type $(3,3,3), (4,4,2)$ and $(6,3,2)$. We
also prove the mirror theorems of Gromov-Witten theory for those orbifolds and
FJRW theory of elliptic singularities. Using T.Milanov and Y. Ruan's work, we
prove the Landau-Ginzburg/Calabi-Yau correspondence of all genera for the above
three types of elliptic orbifold $\mathbb{P}^1$.
