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In this article, we develop and analyze a finite element method with thefirst family N\'ed\'elec elements of the lowest degree for solving a Maxwellinterface problem modeled by a $\mathbf{H}(\text{curl})$-elliptic equation onunfitted meshes. To capture the jump conditions optimally, we construct and use$\mathbf{H}(\text{curl})$ immersed finite element (IFE) functions on interfaceelements while keep using the standard N\'ed\'elec functions on all thenon-interface elements. We establish a few important properties for the IFEfunctions including the unisolvence according to the edge degrees of freedom,the exact sequence relating to the $H^1$ IFE functions and the optimalapproximation capabilities. In order to achieve the optimal convergence rates,we employ a Petrov-Galerkin method in which the IFE functions are only used asthe trial functions and the standard N\'ed\'elec functions are used as the testfunctions which can eliminate the non-conformity errors. We analyze the inf-supconditions under certain conditions and show the optimal convergence rateswhich are also validated by numerical experiments.

Solving Two Dimensional H(curl)-elliptic Interface Systems with Optimal Convergence On Unfitted Meshes