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Optimal convergence for adaptive IGA boundary element methods for weakly-singular integral equations.

Michael Feischl1, Gregor Gantner2, Alexander Haberl2

  • 1The Red Centre, School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052 Australia.

Numerische Mathematik
|June 16, 2017
PubMed
Summary
This summary is machine-generated.

This study mathematically proves convergence for an adaptive algorithm in isogeometric boundary element methods. The adaptive algorithm refines meshes and knot multiplicity, achieving optimal algebraic rates for enhanced computational accuracy.

Keywords:
65D0765N3865N5065Y20

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Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Boundary Element Methods

Background:

  • Weighted-residual error estimators are crucial for adaptive isogeometric boundary element methods (IGABEM).
  • Previous work proposed an adaptive algorithm for 2D IGABEM, controlling mesh refinement and knot multiplicity.

Purpose of the Study:

  • To provide a rigorous mathematical proof of convergence for the proposed adaptive algorithm.
  • To demonstrate that the algorithm achieves optimal algebraic rates of convergence.

Main Methods:

  • Development of a novel mesh-size function incorporating knot multiplicity.
  • Derivation of inverse estimates for NURBS (Non-Uniform Rational B-Splines) in fractional-order Sobolev norms.
  • Mathematical analysis of the adaptive algorithm's convergence properties.

Main Results:

  • A formal proof establishes that the adaptive algorithm guarantees convergence.
  • The algorithm achieves optimal algebraic rates, signifying high efficiency.
  • The novel mesh-size function effectively manages both mesh and knot refinement.

Conclusions:

  • The adaptive algorithm for IGABEM is mathematically validated for convergence with optimal rates.
  • This work advances the theoretical foundation of adaptive methods in isogeometric analysis.
  • The findings support the practical application of IGABEM for complex engineering problems.