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Axioms of adaptivity.

C Carstensen1, M Feischl2, M Page2

  • 1Institut für Mathematik, Humboldt Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany ; Department of Computational Science and Engineering, Yonsei University, 120-749 Seoul, Republic of Korea.

Computers & Mathematics with Applications (Oxford, England : 1987)
|May 19, 2015
PubMed
Summary

This study presents four axioms for optimal convergence rates in adaptive finite element methods (AFEMs). It refines AFEMs by addressing inexact solvers and boundary data, improving error estimator analysis.

Keywords:
A posteriori error estimatorsBoundary element methodFinite element methodIterative solversLocal mesh-refinementOptimal convergence rates

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Scientific Computing

Background:

  • Adaptive finite element methods (AFEMs) are crucial for solving complex problems efficiently.
  • Existing literature on AFEM optimality often requires specific assumptions like estimator efficiency.
  • Challenges include handling inexact solvers, non-homogeneous data, and discrete lower bounds.

Purpose of the Study:

  • To provide a unified axiomatic framework for proving optimal convergence rates in AFEMs.
  • To refine existing AFEM analyses by relaxing stringent assumptions.
  • To generalize the theory to include inexact solvers and various boundary conditions.

Main Methods:

  • Development of a novel axiomatic approach based on four core axioms.
  • Abstract analysis applicable to both linear and nonlinear problems, independent of specific methods.
  • Investigation of quasi-Galerkin orthogonality and its necessity for convergence.

Main Results:

  • Four axioms are sufficient to guarantee optimality of error estimators in AFEMs.
  • Demonstration that estimator efficiency is not required for proving convergence or quasi-optimal behavior.
  • Establishment of quasi-Galerkin orthogonality as both sufficient and necessary for convergence.
  • The framework accommodates equivalent error estimators, inexact solvers, and non-homogeneous boundary data.

Conclusions:

  • The proposed axiomatic framework unifies and advances the theory of optimal AFEMs.
  • The findings relax key assumptions, broadening the applicability of AFEMs.
  • This work provides a robust foundation for analyzing and developing more efficient adaptive solvers.