Plain convergence of adaptive algorithms without exploiting reliability and efficiency

We consider h-adaptive algorithms in the context of the finite element method and the boundary element method. 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 relies only on structural properties of the estimator, namely stability on nonrefined elements and reduction on refined elements. In particular, the new framework thus also covers problems involving nonlocal operators like the fractional Laplacian or boundary integral equations, where (discrete) efficiency is (currently) not available.