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High Order Unfitted Finite Element Methods for Interface Problems and PDEs on Surfaces

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Transport Processes at Fluidic Interfaces

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

Abstract

In this contribution we treat a special class of recently developed higher order unfitted finite element methods for the discretization of mass and surfactant transport equations. To achieve higher order accuracy for such PDEs on stationary geometries we combine standard techniques for numerical integration and a discretization with a special mesh transformation. This results in a new class of isoparametric unfitted finite element methods. For the treatment of such PDEs on evolving geometries we apply space-time variational formulations. These unfitted finite element techniques result in robust and accurate discretization methods for mass and surfactant transport problems in realistic two-phase flow simulations based on a sharp interface formulation. We present these finite element discretization methods, give theoretical error bounds for classes of model problems and present results of numerical simulations both for such model problems and for challenging two-phase flow applications.

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Acknowledgements

The authors gratefully acknowledge funding by the German Science Foundation (DFG) within the Priority Program (SPP) 1506 “Transport Processes at Fluidic Interfaces”. Furthermore, we thank Jörg Grande for his contributions related to the research on surface PDEs and Yuanjun Zhang for making the results in Sect. 2.4.2 available to us.

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Authors and Affiliations

  1. Institut für Numerische und Angewandte Mathematik, University of Göttingen, 37083, Göttingen, Germany

    Christoph Lehrenfeld

  2. Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, 52056, Aachen, Germany

    Arnold Reusken

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  1. Christoph Lehrenfeld
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Correspondence to Arnold Reusken .

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Editors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt , Darmstadt, Germany

    Dieter Bothe

  2. Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany

    Arnold Reusken

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Lehrenfeld, C., Reusken, A. (2017). High Order Unfitted Finite Element Methods for Interface Problems and PDEs on Surfaces. In: Bothe, D., Reusken, A. (eds) Transport Processes at Fluidic Interfaces. Advances in Mathematical Fluid Mechanics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-56602-3_2

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