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Axioms of adaptivity.

Carstensen C, Feischl M, Page M, Praetorius D - Comput Math Appl (2014)

Bottom Line: Second, efficiency of the error estimator is neither needed to prove convergence nor quasi-optimal convergence behavior of the error estimator.Third, some general quasi-Galerkin orthogonality is not only sufficient, but also necessary for the [Formula: see text]-linear convergence of the error estimator, which is a fundamental ingredient in the current quasi-optimality analysis due to Stevenson 2007.Finally, the general analysis allows for equivalent error estimators and inexact solvers as well as different non-homogeneous and mixed boundary conditions.

View Article: PubMed Central - PubMed

Affiliation: Institut 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.

ABSTRACT

This paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of (discrete) lower bounds, inexact solvers, inhomogeneous boundary data, or the use of equivalent error estimators. Solely four axioms guarantee the optimality in terms of the error estimators. Compared to the state of the art in the temporary literature, the improvements of this article can be summarized as follows: First, a general framework is presented which covers the existing literature on optimality of adaptive schemes. The abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method. Second, efficiency of the error estimator is neither needed to prove convergence nor quasi-optimal convergence behavior of the error estimator. In this paper, efficiency exclusively characterizes the approximation classes involved in terms of the best-approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant. Third, some general quasi-Galerkin orthogonality is not only sufficient, but also necessary for the [Formula: see text]-linear convergence of the error estimator, which is a fundamental ingredient in the current quasi-optimality analysis due to Stevenson 2007. Finally, the general analysis allows for equivalent error estimators and inexact solvers as well as different non-homogeneous and mixed boundary conditions.

No MeSH data available.


Related in: MedlinePlus

Map of the quasi-optimality proof. The arrows mark the dependences of the arguments.
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f000005: Map of the quasi-optimality proof. The arrows mark the dependences of the arguments.

Mentions: The following sections are dedicated to the relations between the different axioms and the corresponding implications. Fig. 1 outlines the convergence and quasi-optimality proof in Section  4 and visualizes how the different axioms interact.


Axioms of adaptivity.

Carstensen C, Feischl M, Page M, Praetorius D - Comput Math Appl (2014)

Map of the quasi-optimality proof. The arrows mark the dependences of the arguments.
© Copyright Policy - CC BY-NC-ND
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4384743&req=5

f000005: Map of the quasi-optimality proof. The arrows mark the dependences of the arguments.
Mentions: The following sections are dedicated to the relations between the different axioms and the corresponding implications. Fig. 1 outlines the convergence and quasi-optimality proof in Section  4 and visualizes how the different axioms interact.

Bottom Line: Second, efficiency of the error estimator is neither needed to prove convergence nor quasi-optimal convergence behavior of the error estimator.Third, some general quasi-Galerkin orthogonality is not only sufficient, but also necessary for the [Formula: see text]-linear convergence of the error estimator, which is a fundamental ingredient in the current quasi-optimality analysis due to Stevenson 2007.Finally, the general analysis allows for equivalent error estimators and inexact solvers as well as different non-homogeneous and mixed boundary conditions.

View Article: PubMed Central - PubMed

Affiliation: Institut 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.

ABSTRACT

This paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of (discrete) lower bounds, inexact solvers, inhomogeneous boundary data, or the use of equivalent error estimators. Solely four axioms guarantee the optimality in terms of the error estimators. Compared to the state of the art in the temporary literature, the improvements of this article can be summarized as follows: First, a general framework is presented which covers the existing literature on optimality of adaptive schemes. The abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method. Second, efficiency of the error estimator is neither needed to prove convergence nor quasi-optimal convergence behavior of the error estimator. In this paper, efficiency exclusively characterizes the approximation classes involved in terms of the best-approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant. Third, some general quasi-Galerkin orthogonality is not only sufficient, but also necessary for the [Formula: see text]-linear convergence of the error estimator, which is a fundamental ingredient in the current quasi-optimality analysis due to Stevenson 2007. Finally, the general analysis allows for equivalent error estimators and inexact solvers as well as different non-homogeneous and mixed boundary conditions.

No MeSH data available.


Related in: MedlinePlus