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

Bertrand, Fleurianne; Boffi, Daniele; G. de Diego, Gonzalo

doi:10.1007/s10444-021-09852-z

Open Access
Published: 19 April 2021

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

Fleurianne Bertrand^1,2,3,
Daniele Boffi^4,5 &
Gonzalo G. de Diego ORCID: orcid.org/0000-0002-9896-024X⁶

Advances in Computational Mathematics volume 47, Article number: 34 (2021) Cite this article

270 Accesses
1 Altmetric
Metrics details

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.

References

Gravenkamp, H., Saputra, A.A., Song, C., Birk, C.: Efficient wave propagation simulation on quadtree meshes using SBFEM with reduced modal basis. Int. J. Numer. Methods Eng. 110(12), 1119–1141 (2016)
MathSciNet Article Google Scholar
Kufner, A.: Weighted Sobolev Spaces. Teubner-Texte zur Mathematik. B.G. Teubner, Berlin (1985)
Google Scholar
Liu, L., Zhang, J., Song, C., Birk, C., Saputra, A.A., Gao, W.: Automatic three-dimensional acoustic-structure interaction analysis using the scaled boundary finite element method. J. Comput. Phys. 395, 432–460 (2019)
MathSciNet Article Google Scholar
Ooi, E.T., Song, C., Tin-Loi, F., Yang, Z.: Polygon scaled boundary finite elements for crack propagation modelling. Int. J. Numer. Methods Eng. 91(3), 319–342 (2012)
MathSciNet Article Google Scholar
Ooi, E.T., Song, C., Tin-Loi, F.: A scaled boundary polygon formulation for elasto-plastic analyses. Comput. Methods Appl. Mech. Eng. 268, 905–937 (2014)
MathSciNet Article Google Scholar
Song, C., Wolf, J.P.: The scaled boundary finite-element method-alias consistent infinitesimal finite-element cell method-for elastodynamics. Comput. Methods Appl. Mech. Eng. 147(3), 329–355 (1997)
MathSciNet Article Google Scholar
Song, C., Wolf, J.P.: The scaled boundary finite element method–alias consistent infinitesimal finite element cell method–for diffusion. Int. J. Numer. Methods Eng. 45(10), 1403–1431 (1999)
Article Google Scholar
Song, C., Wolf, J.P.: Semi-analytical representation of stress singularities as occurring in cracks in anisotropic multi-materials with the scaled boundary finite-element method. Comput. Struct. 80(2), 183–197 (2002)
Article Google Scholar
Wolf, J.: Scaled boundary finite element method. Wiley, New York (2003)
Google Scholar
Ziemer, W.P.: Weakly differentiable functions. Springer, Berlin (1989)
Book Google Scholar

Download references

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.

Author information

Authors and Affiliations

Humboldt-Universität zu Berlin, Berlin, Germany
Fleurianne Bertrand
King Abdullah University of Science and Technology, Jeddah, Saudi Arabia
Fleurianne Bertrand
University of Twente, Enschede, the Netherlands
Fleurianne Bertrand
King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia
Daniele Boffi
University of Pavia, Pavia, Italy
Daniele Boffi
Mathematical Institute, University of Oxford, Oxford, UK
Gonzalo G. de Diego

Authors

Fleurianne Bertrand
View author publications
You can also search for this author in PubMed Google Scholar
Daniele Boffi
View author publications
You can also search for this author in PubMed Google Scholar
Gonzalo G. de Diego
View author publications
You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gonzalo G. de Diego.

Additional information

Communicated by: Jon Wilkening

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Cite this article

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

Download citation

Received: 04 May 2020
Accepted: 08 February 2021
Published: 19 April 2021
DOI: https://doi.org/10.1007/s10444-021-09852-z

Keywords

Scaled boundary finite element method
Error analysis
Singular solutions

Mathematics Subject Classification (2010)

65N12
65N15
65N30
65N38