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

Hiptmair, R. ; Li, J. ; Zou, J.

In: Journal of Numerical Mathematics, 2010, vol. 18, no. 3, p. 187-218

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    Titre
    • Convergence analysis of finite element methods for H(div;Ω)-elliptic interface problems
    Auteur
    • Hiptmair, R.. SAM, ETH, Zürich, CH-8092 Zürich, Switzerland. Email: hiptmair@sam.math.ethz.ch
    • Li, J.. SAM, ETH, Zürich, CH-8092 Zürich, Switzerland. Email: jingzhi.li@sam.math.ethz.ch
    • Zou, J.. SAM, ETH, Zürich, CH-8092 Zürich, Switzerland. Email: zou@math.cuhk.edu.hk
    Type de document
    • Postprint
    Langue
    • Anglais
    Publié dans
    • Journal of Numerical Mathematics, 2010, vol. 18, no. 3, p. 187-218. Walter de Gruyter GmbH & Co. KG
    Autre version électronique
    Identifiant OAI-PMH
    • oai:doc.rero.ch:297573
    Summary
    In this article we analyze a finite element method for solving H(div;Ω)-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;Ω)-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 δ that quantifies the mismatch between the smooth interface and the finite element mesh. Optimal error estimates in the H(div;Ω)-norms are obtained for the first time. The analysis is based on a so-called δ-strip argument, a new extension theorem for H 1(div)-functions across smooth interfaces, a novel non-standard interfaceaware interpolation operator, and a perturbation argument for degrees of freedom in H(div;Ω)-conforming finite elements. Numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution