The finite element method for solving second-order both elliptic and parabolic interface problems is studied. It is proved that the method converges as the usual non-interface elliptic and parabolic problems, both for the energy-norm and the \(L\)-norm. The resultant linear systems are always symmetric and positive definite when the original partial differential equations are self-adjoint and uniformly elliptic. The authors approximate the smooth interface by a polygon, and the interface function by its interpolant. Here the approximation problem seems similar to the classical finite element method.