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Trace Finite Element Methods for PDEs on Surfaces

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Geometrically Unfitted Finite Element Methods and Applications

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 121))

Abstract

In this paper we consider a class of unfitted finite element methods for discretization of partial differential equations on surfaces. In this class of methods known as the Trace Finite Element Method (TraceFEM), restrictions or traces of background surface-independent finite element functions are used to approximate the solution of a PDE on a surface. We treat equations on steady and time-dependent (evolving) surfaces. Higher order TraceFEM is explained in detail. We review the error analysis and algebraic properties of the method. The paper navigates through the known variants of the TraceFEM and the literature on the subject.

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Acknowledgements

The authors acknowledge the contributions of A. Chernyshenko, A. Demlow, J. Grande, S. Gross, C. Lehrenfeld, and X. Xu to the research topics treated in this article. M. Olshanskii was partially supported by MSF through the Division of Mathematical Sciences grant 1717516.

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Authors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, TX, 77204-3008, USA

    Maxim A. Olshanskii

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

    Arnold Reusken

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  1. Maxim A. Olshanskii
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Correspondence to Maxim A. Olshanskii .

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Editors and Affiliations

  1. Faculté des Sciences, de la Technologie et de la Communication, Université du Luxembourg, Luxembourg

    Stéphane P. A. Bordas

  2. Department of Mathematics, University College London, United Kingdom

    Erik Burman

  3. Department of Mathematics and Mathematical Statistics, Umeå University, Sweden

    Mats G. Larson

  4. Department of Mathematics, University of Houston, USA

    Maxim A. Olshanskii

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Olshanskii, M.A., Reusken, A. (2017). Trace Finite Element Methods for PDEs on Surfaces. In: Bordas, S., Burman, E., Larson, M., Olshanskii, M. (eds) Geometrically Unfitted Finite Element Methods and Applications. Lecture Notes in Computational Science and Engineering, vol 121. Springer, Cham. https://doi.org/10.1007/978-3-319-71431-8_7

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