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ZZ-Type a posteriori error estimators for adaptive boundary element methods on a curve.
Michael Feischl1, Thomas Führer1, Michael Karkulik2
1Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8-10/E101/4, 1040 Wien, Austria.
This study introduces ZZ-type error estimators for the adaptive boundary element method (BEM), extending well-established techniques from adaptive finite element methods (FEM). Convergence of adaptive mesh-refining algorithms is proven for integral equations.
Area of Science:
- Computational Mathematics
- Numerical Analysis
- Boundary Element Methods
Background:
- ZZ-error estimators are established for adaptive finite element methods (FEM).
- Adaptive boundary element methods (BEM) require robust error estimation for mesh refinement.
- Existing error estimators for BEM may not fully leverage ZZ-type approaches.
Purpose of the Study:
- To propose and analyze ZZ-type error estimators specifically for the adaptive boundary element method (BEM).
- To extend the mathematical framework of ZZ-error estimators to integral equations.
- To demonstrate the convergence of adaptive mesh-refining algorithms utilizing these new estimators.
Main Methods:
- Development of ZZ-type error estimators tailored for BEM formulations.
- Mathematical analysis of the proposed estimators for weakly singular and hyper-singular integral equations.
- Implementation and numerical verification of the adaptive BEM algorithms.
Main Results:
- Successful proposal and analysis of ZZ-type error estimators for adaptive BEM.
- Proof of convergence for adaptive mesh-refining algorithms employing these estimators.
- Validation of theoretical findings through numerical experiments.
Conclusions:
- ZZ-type error estimators are effective for adaptive BEM.
- The proposed estimators enhance the reliability of adaptive mesh refinement in BEM.
- This work provides a strong theoretical and numerical foundation for ZZ-type estimators in BEM.
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