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The solver tackles a linear system which is generated by the discretization of a second-order elliptic diffusion problem using conforming finite elements of polynomial order<jats:inline-formula id=\"j_cmam-2020-0024_ineq_9999\"><jats:alternatives><m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><m:mrow><m:mi>p<\/m:mi><m:mo>\u2265<\/m:mo><m:mn>1<\/m:mn><\/m:mrow><\/m:math><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2020-0024_eq_0320.png\"\/><jats:tex-math>{p\\geq 1}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. After one V-cycle (\u201cfull-smoothing\u201d substep) of the solver of [A. Mira\u00e7i, J. Pape\u017e, and M. Vohral\u00edk, A-posteriori-steered<jats:italic>p<\/jats:italic>-robust multigrid with optimal step-sizes and adaptive number of smoothing steps, SIAM J. Sci. Comput. 2021, 10.1137\/20M1349503], we dispose of a reliable, efficient, and localized estimation of the algebraic error. We use this existing result to develop our new adaptive algorithm: thanks to the information of the estimator and based on a bulk-chasing criterion, cf. [W. D\u00f6rfler, A convergent adaptive algorithm for Poisson\u2019s equation, SIAM J. Numer. Anal. 33 1996, 3, 1106\u20131124], we mark patches of elements with increased estimated error on all levels. Then, we proceed by a modified and cheaper V-cycle (\u201cadaptive-smoothing\u201d substep), which only applies smoothing in the marked regions. The proposed adaptive multigrid solver picks autonomously and adaptively the optimal step-size per level as in our previous work but also the type of smoothing per level (weighted restricted additive or additive Schwarz) and concentrates smoothing to marked regions with high error. We prove that, under a numerical condition that we verify in the algorithm, each substep (full and adaptive) contracts the error<jats:italic>p<\/jats:italic>-robustly, which is confirmed by numerical experiments. Moreover, the proposed algorithm behaves numerically robustly with respect to the number of levels as well as to the diffusion coefficient jump for a uniformly-refined hierarchy of meshes.<\/jats:p>","DOI":"10.1515\/cmam-2020-0024","type":"journal-article","created":{"date-parts":[[2021,2,6]],"date-time":"2021-02-06T00:06:35Z","timestamp":1612569995000},"page":"445-468","source":"Crossref","is-referenced-by-count":1,"title":["Contractive Local Adaptive Smoothing Based on D\u00f6rfler\u2019s Marking in A-Posteriori-Steered<i>p<\/i>-Robust Multigrid Solvers"],"prefix":"10.1515","volume":"21","author":[{"given":"Ani","family":"Mira\u00e7i","sequence":"first","affiliation":[{"name":"Inria , 2 rue Simone Iff, 75589 Paris ; and CERMICS, Ecole des Ponts, 77455 Marne-la-Vall\u00e9e , France"}]},{"given":"Jan","family":"Pape\u017e","sequence":"additional","affiliation":[{"name":"Institute of Mathematics , Czech Academy of Sciences , \u017ditn\u00e1 25, 115\u200967 Prague , Czech Republic"}]},{"given":"Martin","family":"Vohral\u00edk","sequence":"additional","affiliation":[{"name":"Inria , 2 rue Simone Iff, 75589 Paris ; and CERMICS, Ecole des Ponts, 77455 Marne-la-Vall\u00e9e , France"}]}],"member":"374","published-online":{"date-parts":[[2021,2,5]]},"reference":[{"key":"2023033111200647653_j_cmam-2020-0024_ref_001","doi-asserted-by":"crossref","unstructured":"A. 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