  A simple and efficient interface-fitted mesh generation algorithm is
developed in this paper. This algorithm can produce a local anisotropic fitting
mixed mesh which consists of both triangles and quadrilaterals near the
interface. A new finite element method is proposed for second order elliptic
interface problems based on the resulting mesh. Optimal approximation
capabilities on anisotropic elements are proved in both the $H^1$ and $L^2$
norms. The discrete system is usually ill-conditioned due to anisotropic and
small elements near the interface. Thereupon, a multigrid method is presented
to handle this issue. The convergence rate of the multigrid method is shown to
be optimal with respect to both the coefficient jump ratio and mesh size.
Numerical experiments are presented to demonstrate the theoretical results.
