  New finite element methods are proposed for elliptic interface problems in
one and two dimensions. The main motivation is not only to get an accurate
solution but also an accurate first order derivative at the interface (from
each side). The key in 1D is to use the idea from \cite{wheeler1974galerkin}.
For 2D interface problems, the idea is to introduce a small tube near the
interface and introduce the gradient as part of unknowns, which is similar to a
mixed finite element method, except only at the interface. Thus the
computational cost is just slightly higher than the standard finite element
method. We present rigorous one dimensional analysis, which show second order
convergence order for both of the solution and the gradient in 1D. For two
dimensional problems, we present numerical results and observe second order
convergence for the solution, and super-convergence for the gradient at the
interface.
