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

TL;DR

This paper introduces a finite element method using Nédélec and immersed finite element functions to solve 2D H(curl)-elliptic interface problems on unfitted meshes, achieving optimal convergence rates validated by analysis and experiments.

Contribution

It develops a novel Petrov-Galerkin finite element approach with IFE functions for Maxwell interface problems on unfitted meshes, ensuring optimal convergence.

Findings

Achieves optimal convergence rates for the proposed method.

Validates theoretical results with numerical experiments.

Establishes properties of IFE functions including unisolvence and approximation capabilities.

Abstract

In this article, we develop and analyze a finite element method with the first family N\'ed\'elec elements of the lowest degree for solving a Maxwell interface problem modeled by a $\mathbf{H}(\text{curl})$-elliptic equation on unfitted meshes. To capture the jump conditions optimally, we construct and use $\mathbf{H}(\text{curl})$ immersed finite element (IFE) functions on interface elements while keep using the standard N\'ed\'elec 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 N\'ed\'elec 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.