We analyze an immersed interface finite element method based on linear polynomials on non-interface triangular elements and piecewise linear polynomials on interface triangular elements. The flux jump condition is weakly enforced on the smooth interface. Error estimates are derived in the broken $H^1$-norm and $L^2$-norm.
And we cunstruct and analyze a nonconforming immersed interface finite element method with Cartesian triangular grids using piecewise linear basis functions around interface having degrees of freedom on midpoints of edges and Petrov-Galerkin method using the standard linear functions as test functions and derive an error estimates. We also give numerical results for the schemes, which show the optimal order of convergence of the error.