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Estimator reduction and convergence of adaptive BEM
Markus Aurada1, Samuel Ferraz-Leite, Dirk Praetorius
1Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstr. 8-10, 1040 Wien, Austria.
Applied Numerical Mathematics : Transactions of IMACS
|March 14, 2013
Summary
This study introduces a new convergence concept for adaptive boundary element methods, focusing on estimator convergence rather than direct error reduction. This approach provides mathematical backing for anisotropic mesh refinement in 3D computations.
Area of Science:
- Scientific Computing
- Numerical Analysis
- Computational Mathematics
Background:
- Adaptive mesh-refining algorithms are crucial in scientific computing.
- Convergence analysis for adaptive boundary element schemes remains an open challenge, unlike adaptive finite element methods.
Purpose of the Study:
- To propose and analyze a relaxed notion of convergence for adaptive boundary element schemes.
- To demonstrate that estimator convergence is achievable and sufficient for adaptive algorithms.
- To provide mathematical justification for anisotropic mesh refinement in 3D boundary element computations.
Main Methods:
- Developing a relaxed convergence criterion based on estimator convergence.
- Utilizing Dörfler marking and inverse estimates for analysis.
- Focusing on the estimator reduction property of error estimators.
Main Results:
- Established estimator convergence for adaptive boundary element schemes under specific conditions.
- Demonstrated that estimator reduction property is sufficient for estimator convergence.
- Provided the first mathematical justification for anisotropic mesh refinement steering in 3D.
Conclusions:
- The proposed relaxed convergence notion offers a viable path for analyzing adaptive boundary element methods.
- The findings support the use of anisotropic mesh refinement for optimal performance in 3D computations.
- This work opens new avenues for understanding and improving adaptive boundary element schemes.
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