  In recent years, the immersed finite element methods (IFEM) introduced in
\cite{Li2003}, \cite{Li2004} to solve elliptic problems having an interface in
the domain due to the discontinuity of coefficients are getting more attentions
of researchers because of their simplicity and efficiency. Unlike the
conventional finite element methods, the IFEM allows the interface cut through
the interior of the element, yet after the basis functions are altered so that
they satisfy the flux jump conditions, it seems to show a reasonable order of
convergence. In this paper, we propose an improved version of the $P_1$ based
IFEM by adding the line integral of flux terms on each element. This technique
resembles the discontinuous Galerkin (DG) method, however, our method has much
less degrees of freedom than the DG methods since we use the same number of
unknowns as the conventional $P_1$ finite element method. We prove $H^1$ and
$L^2$ error estimates which are optimal both in order and regularity. Numerical
experiments were carried out for several examples, which show the robustness of
our scheme.
