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Convergence analysis of finite element methods for H(div)-elliptic interface problems

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

(Report number 2010-14)

Abstract
In this article we analyze a finite element method for solving H(div;Omega)-elliptic interface problems in general three-dimensional Lipschitz domains with smooth material interfaces. The continuous problems are discretized by means of lowest order H(div;Omega)-conforming finite elements of the first family (Raviart-Thomas or Nédélec face elements) on a family of unstructured oriented tetrahedral meshes. These resolve the smooth interface in the sense of sufficient approximation in terms of a parameter 8 that quantifies the mismatch between the smooth interface and the finite element mesh. Optimal error estimates in the H(div;Omega)-norms are obtained for the first time. The analysis is based on a so-called 8-strip argument, a new extension theorem for H^1 (div)-functions across smooth interfaces, a novel non-standard interface-aware interpolation operator, and a perturbation argument for degrees of freedom in H(div;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(div;Omega)-elliptic interface problems, finite element methods, face elements, convergence analysis

@Techreport{HLZ10_38,
  author = {R. Hiptmair and J.-Z. Li and J. Zou},
  title = {Convergence analysis of finite element methods for H(div)-elliptic interface problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2010-14},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2010/2010-14.pdf },
  year = {2010}
}

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