  We propose a discontinuous Galerkin(DG) method to approximate the elliptic
interface problem on unfitted mesh using a new approximation space. The
approximation space is constructed by patch reconstruction with one degree of
freedom per element. The optimal error estimates in both L2 norm and DG energy
norm are obtained, without the typical constraints for DG method on how the
interface intersects to the elements in the mesh. Other than enjoying the
advantages of DG method, our method may achieve even better efficiency than the
conforming finite element method, as illustrated by numerical examples.
