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For the resulting algorithm, we prove a contraction of the (doubly) inexact iterates after some amount of steps of mesh-refinement\/linearization\/algebraic solver, leading to its linear convergence. Moreover, for usual mesh-refinement rules, we also prove that the overall error decays at the optimal rate with respect to the number of elements (degrees of freedom) added with respect to the initial mesh. Finally, we prove that our fully adaptive algorithm drives the overall error down with the same optimal rate also with respect to the overall algorithmic cost expressed as the cumulated sum of the number of mesh elements over all mesh-refinement, linearization, and algebraic solver steps. Numerical experiments support these theoretical findings and illustrate the optimal overall algorithmic cost of the fully adaptive algorithm on several test cases.<\/jats:p>","DOI":"10.1007\/s00211-021-01176-w","type":"journal-article","created":{"date-parts":[[2021,2,6]],"date-time":"2021-02-06T14:22:40Z","timestamp":1612621360000},"page":"679-725","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["Convergence and quasi-optimal cost of adaptive algorithms for nonlinear operators including iterative linearization and algebraic solver"],"prefix":"10.1007","volume":"147","author":[{"given":"Alexander","family":"Haberl","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dirk","family":"Praetorius","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stefan","family":"Schimanko","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martin","family":"Vohral\u00edk","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,2,5]]},"reference":[{"key":"1176_CR1","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1007\/978-3-319-10705-9_24","volume-title":"Numerical Mathematics and Advanced Applications\u2013ENUMATH 2013","author":"Roland Becker","year":"2015","unstructured":"Becker, Roland, Capatina, Daniela, Luce, Robert: Stopping criteria based on locally reconstructed fluxes. 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