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Convergence analysis of the scaled boundary finite element method for the Laplace equation

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  • Published: 19 April 2021
  • volume 47, Article number: 34 (2021)
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Convergence analysis of the scaled boundary finite element method for the Laplace equation
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  • Fleurianne Bertrand1,2,3,
  • Daniele Boffi4,5 &
  • Gonzalo G. de Diego  ORCID: orcid.org/0000-0002-9896-024X6 

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Abstract

The scaled boundary finite element method (SBFEM) is a relatively recent boundary element method that allows the approximation of solutions to partial differential equations (PDEs) without the need of a fundamental solution. A theoretical framework for the convergence analysis of SBFEM is proposed here. This is achieved by defining a space of semi-discrete functions and constructing an interpolation operator onto this space. We prove error estimates for this interpolation operator and show that optimal convergence to the solution can be obtained in SBFEM. These theoretical results are backed by two numerical examples.

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Acknowledgements

We would like to thank the project partners Prof. Carolin Birk (Universität Duisburg-Essen, Germany) and Prof. Christian Meyer (TU Dortmund, Germany) as well as Professor Gerhard Starke for the fruitful discussions.

Funding

The first author was supported by the German Research Foundation (DFG) in the Priority Programme SPP 1748 Reliable simulation techniques in solid mechanics under grant number BE6511/1-1. The second author is a member of the INdAM Research group GNCS and his research is partially supported by IMATI/CNR and by PRIN/MIUR. The first and third authors were supported by Mercator Research Center Ruhr (MERCUR) under grant Pr-2017-0017.

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Authors and Affiliations

  1. Humboldt-Universität zu Berlin, Berlin, Germany

    Fleurianne Bertrand

  2. King Abdullah University of Science and Technology, Jeddah, Saudi Arabia

    Fleurianne Bertrand

  3. University of Twente, Enschede, the Netherlands

    Fleurianne Bertrand

  4. King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia

    Daniele Boffi

  5. University of Pavia, Pavia, Italy

    Daniele Boffi

  6. Mathematical Institute, University of Oxford, Oxford, UK

    Gonzalo G. de Diego

Authors
  1. Fleurianne Bertrand
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  2. Daniele Boffi
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  3. Gonzalo G. de Diego
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Corresponding author

Correspondence to Gonzalo G. de Diego.

Additional information

Communicated by: Jon Wilkening

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Bertrand, F., Boffi, D. & G. de Diego, G. Convergence analysis of the scaled boundary finite element method for the Laplace equation. Adv Comput Math 47, 34 (2021). https://doi.org/10.1007/s10444-021-09852-z

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  • Received: 04 May 2020

  • Accepted: 08 February 2021

  • Published: 19 April 2021

  • DOI: https://doi.org/10.1007/s10444-021-09852-z

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Keywords

  • Scaled boundary finite element method
  • Error analysis
  • Singular solutions

Mathematics Subject Classification (2010)

  • 65N12
  • 65N15
  • 65N30
  • 65N38

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