G. Gantner, D. Praetorius:

"Plain convergence of adaptive algorithms without exploiting reliability and efficiency";

in: "ASC Report 25/2020", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2020, ISBN: 978-3-902627-13-1, 1 - 17.

We consider h-adaptive algorithms in the context of the finite element method (FEM) and the boundary element method (BEM). Under quite general assumptions on the building blocks SOLVE, ESTIMATE, MARK, and REFINE of such algorithms, we prove plain convergence in the sense that the adaptive algorithm drives the underlying a-posteriori

error estimator to zero. Unlike available results in the literature, our analysis avoids the use of any reliability and efficiency estimate, but only relies on structural properties of

the estimator, namely stability on non-refined elements and reduction on refined elements.

In particular, the new framework thus covers also problems involving non-local operators like the fractional Laplacian or boundary integral equations, where (discrete) efficiency is (currently) not available.

http://www.asc.tuwien.ac.at/preprint/2020/asc25x2020.pdf

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