Abstract
In this article we investigate the analysis of a finite element method for solving H(curl; Ω)-elliptic interface problems in general three-dimensional polyhedral domains with smooth interfaces. The continuous problems are discretized by means of the first family of lowest order Nédélec H(curl; Ω)-conforming finite elements on a family of tetrahedral meshes which resolve the smooth interface in the sense of sufficient approximation in terms of a parameter δ that quantifies the mismatch between the smooth interface and the triangulation. Optimal error estimates in the H(curl; Ω)-norm are obtained for the first time. The analysis is based on a δ-strip argument, a new extension theorem for H 1(curl; Ω)-functions across smooth interfaces, a novel non-standard interface-aware interpolation operator, and a perturbation argument for degrees of freedom for H(curl; Ω)-conforming finite elements. Numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution.
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Hiptmair, R., Li, J. & Zou, J. Convergence analysis of finite element methods for H(curl; Ω)-elliptic interface problems. Numer. Math. 122, 557–578 (2012). https://doi.org/10.1007/s00211-012-0468-6
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DOI: https://doi.org/10.1007/s00211-012-0468-6