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.
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)
Kufner, A.: Weighted Sobolev Spaces. Teubner-Texte zur Mathematik. B.G. Teubner, Berlin (1985)
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)
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)
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)
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)
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)
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)
Wolf, J.: Scaled boundary finite element method. Wiley, New York (2003)
Ziemer, W.P.: Weakly differentiable functions. Springer, Berlin (1989)
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.
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.
Communicated by: Jon Wilkening
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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
- Scaled boundary finite element method
- Error analysis
- Singular solutions
Mathematics Subject Classification (2010)