In this paper, a bilinear Petrov-Galerkin finite element method is introduced to solve the variable matrix coefficient elliptic equation with interfaces using non-body-fitted grid. Different cases the interface cut the cell are discussed. The condition number of the large sparse linear system is studied. Numerical results demonstrate that the method is nearly second order accurate in the $L^\infty$ norm and $L^2$ norm, and is first order accurate in the $H^1$ norm.
Petrov-Galerkin finite element method jump condition bilinear.