  We present a numerical method for two-phase incompressible Navier-Stokes
equation with jump discontinuity in the normal component of the stress tensor
and in the material properties. Although the proposed method is only
first-order accurate, it does capture discontinuity sharply, not neglecting nor
omitting any component of the jump condition. Discontinuities in velocity
gradient and pressure are expressed using a linear combination of singular
force and tangential derivatives of velocities to handle jump conditions in a
fully implicit manner. The linear system for the divergence of the stress
tensor is constructed in the framework of the ghost fluid method, and the
resulting saddle-point system is solved via an iterative procedure. Numerical
results support the inference that the proposed method converges in $L^\infty$
norms even when velocities and pressures are not smooth across the interface
and can handle a large density ratio that is likely to appear in a real-world
simulation.
