Solving Two Dimensional H(curl)-elliptic Interface Systems with Optimal Convergence On Unfitted Meshes
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In this article, we develop and analyze a finite element method with the first family Nédélec elements of the lowest degree for solving a Maxwell interface problem modeled by a 𝐇(curl)-elliptic equation on unfitted meshes. To capture the jump conditions optimally, we construct and use 𝐇(curl) immersed finite element (IFE) functions on interface elements while keep using the standard Nédélec functions on all the non-interface elements. We establish a few important properties for the IFE functions including the unisolvence according to the edge degrees of freedom, the exact sequence relating to the H^1 IFE functions and the optimal approximation capabilities. In order to achieve the optimal convergence rates, we employ a Petrov-Galerkin method in which the IFE functions are only used as the trial functions and the standard Nédélec functions are used as the test functions which can eliminate the non-conformity errors. We analyze the inf-sup conditions under certain conditions and show the optimal convergence rates which are also validated by numerical experiments.
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