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Solving two-dimensional H(curl)-elliptic interface systems with optimal convergence on unfitted meshes

Part of: Miscellaneous topics - Partial differential equations Partial differential equations, boundary value problems

Published online by Cambridge University Press:  05 January 2023

Ruchi Guo Open the ORCID record for Ruchi Guo [Opens in a new window] ,
Yanping Lin  and
Jun Zou
Ruchi Guo*
Affiliation:
Department of Mathematics, University of California, Irvine, CA 92697, USA
Yanping Lin
Affiliation:
Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China
Jun Zou
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
*
*Correspondence author. Email: ruchig@uci.edu
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Abstract

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.

Keywords

Maxwell equationsinterface problemsH(curl)-elliptic equationsNédélec elementsimmersed finite element methodsPetrov–Galerkin formulationexact sequence

MSC classification

Primary: 65N15: Error bounds
Secondary: 35R05: Partial differential equations with discontinuous coefficients or data 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Papers
Information
European Journal of Applied Mathematics , Volume 34 , Issue 4 , August 2023 , pp. 774 - 805
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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