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Solving two-dimensional H(curl)-elliptic interface systems with optimal convergence on unfitted meshes
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- Journal:
- European Journal of Applied Mathematics / Volume 34 / Issue 4 / August 2023
- Published online by Cambridge University Press:
- 05 January 2023, pp. 774-805
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- Article
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Finite element methods developed for unfitted meshes have been widely applied to various interface problems. However, many of them resort to non-conforming spaces for approximation, which is a critical obstacle for the extension to
$\textbf{H}(\text{curl})$ equations. This essential issue stems from the underlying Sobolev space 
$\textbf{H}^s(\text{curl};\,\Omega)$, and even the widely used penalty methodology may not yield the optimal convergence rate. One promising approach to circumvent this issue is to use a conforming test function space, which motivates us to develop a Petrov–Galerkin immersed finite element (PG-IFE) method for 
$\textbf{H}(\text{curl})$-elliptic interface problems. We establish the Nédélec-type IFE spaces and develop some important properties including their edge degrees of freedom, an exact sequence relating to the 
$H^1$ IFE space and optimal approximation capabilities. We analyse the inf-sup condition under certain assumptions and show the optimal convergence rate, which is also validated by numerical experiments.
