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Convergence analysis of finite element methods for h(curl)-elliptic interface problems

by R. Hiptmair and J.-Z. Li and J. Zou

(Report number 2009-04)

Abstract
In this article we analyse a finite element method for solving H(curl;\Omega)-elliptic interface problems in general three-dimensional polyhedral domains with smooth material interfaces. The continuous problems are discretized by means of the first family of lowest order Nédélec H(curl;\Omega)-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 \delta that quantifies the mismatch between the smooth interface and the triangulation. Optimal error estimates in the H(curl;\Omega) norm are obtained for the first time. The analysis is based on a so-called \delta-strip argument, a new extension theorem for H1(curl)-functions across smooth interfaces, a novel non standard interface-aware interpolation operator, and a perturbation argument for degrees of freedomforH(curl; Omega)-conforming finite elements. Numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution.

Keywords: H(curl;\Omega)-elliptic interface problems, finite element methods, Nédélec's edge elements, convergence analysis

@Techreport{HLZ09_66,
  author = {R. Hiptmair and J.-Z. Li and J. Zou},
  title = {Convergence analysis of finite element methods for h(curl)-elliptic interface problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2009-04},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-04.pdf },
  year = {2009}
}

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