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Approximate Integration01:24

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In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...
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Reliable and efficient a posteriori error estimation for adaptive IGA boundary element methods for weakly-singular

Michael Feischl1, Gregor Gantner1, Dirk Praetorius1

  • 1Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstraße 8-10, A-1040 Wien, Austria.

Computer Methods in Applied Mechanics and Engineering
|June 19, 2015
PubMed
Summary

This study introduces residual-type a posteriori error estimators for the Galerkin boundary element method (BEM) in 2D. The adaptive isogeometric BEM algorithm achieves optimal convergence for weakly-singular integral equations.

Keywords:
A posteriori error estimateAdaptive mesh-refinementBoundary element methodIsogeometric analysis

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Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Boundary Element Methods

Background:

  • Weakly-singular integral equations of the first kind are common in 2D boundary element method (BEM) analysis.
  • Existing error estimators often require strong assumptions on boundary parametrization and mesh properties.

Purpose of the Study:

  • To develop and analyze residual-type a posteriori error estimators for Galerkin BEM in 2D.
  • To provide both lower and upper bounds for the Galerkin BEM error.
  • To contribute to adaptive BEM within isogeometric analysis (IGABEM).

Main Methods:

  • Analysis of residual-type a posteriori error estimators.
  • Formulation of an adaptive algorithm for isogeometric BEM (IGABEM).
  • Utilizing piecewise smooth parametrizations, local mesh-refinement, and NURBS.

Main Results:

  • The proposed error estimators provide reliable lower and upper bounds for the Galerkin BEM error.
  • The analysis supports piecewise smooth parametrizations and local mesh-refinement.
  • The developed adaptive IGABEM algorithm effectively steers refinement and knot multiplicity.

Conclusions:

  • The adaptive IGABEM strategy, guided by the new error estimators, demonstrates optimal convergence rates.
  • This work provides a theoretical foundation for adaptive BEM in IGABEM.
  • Numerical experiments validate the effectiveness of the proposed adaptive approach.