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

翻译：本文发现了一个关于使用边积分值作为自由度的非协调浸入式有限元方法（IFE）求解椭圆界面问题的重要结论。我们表明，当精确解的切向导数及系数跳跃在界面上非零时，此类无惩罚项的非协调IFE方法无法保证最优收敛性。我们提供了一个非平凡的反例来佐证理论分析。为恢复最优收敛率，我们发展了一种新型非协调IFE方法，其在界面边上引入局部附加项。该新方法无需参数，消除了传统部分惩罚IFE方法的局限性。我们证明了IFE基函数在任意三角形上的唯一可解性，该性质在文献中尚未被讨论。此外，区别于多点泰勒展开，我们通过统一方法推导了Crouzeix-Raviart与旋转$Q_1$ IFE空间的最优逼近能力，该方法可轻松处理变系数情形。最后，我们证明了$H^1$范数与$L^2$范数下的最优误差估计，并通过数值实验进行了验证。