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

Fleurianne Bertrand; Daniele Boffi; Gonzalo G. de Diego

doi:10.1007/s10444-021-09852-z

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

Fleurianne Bertrand
, Daniele Boffi
, Gonzalo G. de Diego^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

3 Citations (Scopus)

46 Downloads (Pure)

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.

Original language	English
Article number	34
Number of pages	17
Journal	Advances in computational mathematics
Volume	47
Issue number	3
Early online date	19 Apr 2021
DOIs	https://doi.org/10.1007/s10444-021-09852-z
Publication status	Published - Jun 2021

Keywords

Error analysis
Scaled boundary finite element method
Singular solutions

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10.1007/s10444-021-09852-zLicence: CC BY

Bertrand2021_Article_ConvergenceAnalysisOfTheScaledFinal published version, 794 KBLicence: CC BY

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title = "Convergence analysis of the scaled boundary finite element method for the Laplace equation",

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.",

keywords = "Error analysis, Scaled boundary finite element method, Singular solutions",

author = "Fleurianne Bertrand and Daniele Boffi and \{G. de Diego\}, Gonzalo",

note = "Funding Information: 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. Funding Information: We would like to thank the project partners Prof. Carolin Birk (Universit?t Duisburg-Essen, Germany) and Prof. Christian Meyer (TU Dortmund, Germany) as well as Professor Gerhard Starke for the fruitful discussions. Publisher Copyright: {\textcopyright} 2021, The Author(s).",

year = "2021",

month = jun,

doi = "10.1007/s10444-021-09852-z",

language = "English",

volume = "47",

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AU - Bertrand, Fleurianne

AU - Boffi, Daniele

AU - G. de Diego, Gonzalo

N1 - Funding Information: 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. Funding Information: We would like to thank the project partners Prof. Carolin Birk (Universit?t Duisburg-Essen, Germany) and Prof. Christian Meyer (TU Dortmund, Germany) as well as Professor Gerhard Starke for the fruitful discussions. Publisher Copyright: © 2021, The Author(s).

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Y1 - 2021/6

N2 - 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.

AB - 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.

KW - Error analysis

KW - Scaled boundary finite element method

KW - Singular solutions

UR - https://www.scopus.com/pages/publications/85104664417

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DO - 10.1007/s10444-021-09852-z

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VL - 47

JO - Advances in computational mathematics

JF - Advances in computational mathematics

IS - 3

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ER -

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

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Keywords

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

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