Convergence of simple adaptive Galerkin schemes based on h − h/2 error estimators
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We discuss several adaptive mesh-refinement strategies based on (h − h/2)-error estimation. This class of adaptive methods is particularly popular in practise since it is problem independent and requires virtually no implementational overhead. We prove that, under the saturation assumption, these adaptive algorithms are convergent. Our framework applies not only to finite element methods, but also yields a first convergence proof for adaptive boundary element schemes. For a finite element model problem, we extend the proposed adaptive scheme and prove convergence even if the saturation assumption fails to hold in general.
Mathematics Subject Classification (2000)65N30 65N38 65N50 65N12
Parts of the results have been achieved during a research stay of C.O. and D.P. at the Hausdorff Institute for Mathematics in Bonn, which is thankfully acknowledged. S.F. acknowledges a grant of the graduate school Differential Equations – Models in Science and Engineering, funded by the Austrian Science Fund (FWF) under grant W800-N05. D.P. is partially supported through the research project Adaptive Boundary Element Method, funded by the Austrian Science Fund (FWF) under grant P21732.
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