  We study some numerical methods for solving second order elliptic problem
with interface. We introduce an immersed interface finite element method based
on the `broken' $P_1$-nonconforming piecewise linear polynomials on interface
triangular elements having edge averages as degrees of freedom. This linear
polynomials are broken to match the homogeneous jump condition along the
interface which is allowed to cut through the element. We prove optimal orders
of convergence in $H^1$ and $L^2$-norm. Next we propose a mixed finite volume
method in the context introduced in \cite{Kwak2003} using the Raviart-Thomas
mixed finite element and this `broken' $P_1$-nonconforming element. The
advantage of this mixed finite volume method is that once we solve the
symmetric positive definite pressure equation(without Lagrangian multiplier),
the velocity can be computed locally by a simple formula. This procedure avoids
solving the saddle point problem. Furthermore, we show optimal error estimates
of velocity and pressure in our mixed finite volume method. Numerical results
show optimal orders of error in $L^2$-norm and broken $H^1$-norm for the
pressure, and in $H(\Div)$-norm for the velocity.
