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In this paper, an important discovery has been found for nonconformingimmersed finite element (IFE) methods using the integral values on edges asdegrees of freedom for solving elliptic interface problems. We show that thoseIFE methods without penalties are not guaranteed to converge optimally if thetangential derivative of the exact solution and the jump of the coefficient arenot zero on the interface. A nontrivial counter example is also provided tosupport our theoretical analysis. To recover the optimal convergence rates, wedevelop a new nonconforming IFE method with additional terms locally oninterface edges. The new method is parameter-free which removes the limitationof the conventional partially penalized IFE method. We show the IFE basisfunctions are unisolvent on arbitrary triangles which is not considered in theliterature. Furthermore, different from multipoint Taylor expansions, we derivethe optimal approximation capabilities of both the Crouzeix-Raviart and therotated-$Q_1$ IFE spaces via a unified approach which can handle the case ofvariable coefficients easily. Finally, optimal error estimates in both $H^1$-and $L^2$- norms are proved and confirmed with numerical experiments.

Analysis of nonconforming IFE methods and a new scheme for elliptic interface problems