{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T11:15:08Z","timestamp":1762254908337,"version":"3.40.5"},"reference-count":53,"publisher":"Walter de Gruyter GmbH","issue":"2","license":[{"start":{"date-parts":[[2024,6,26]],"date-time":"2024-06-26T00:00:00Z","timestamp":1719360000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,4,1]]},"abstract":"Abstract<\/jats:title>\n The p<\/jats:italic>-Laplacian problem \n \n \n \n \n -<\/m:mo>\n \n \u2207<\/m:mo>\n \u22c5<\/m:mo>\n \n (<\/m:mo>\n \n \n (<\/m:mo>\n \n \u03bc<\/m:mi>\n +<\/m:mo>\n \n \n |<\/m:mo>\n \n \u2207<\/m:mo>\n \u2061<\/m:mo>\n u<\/m:mi>\n <\/m:mrow>\n |<\/m:mo>\n <\/m:mrow>\n \n p<\/m:mi>\n -<\/m:mo>\n 2<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n \u2062<\/m:mo>\n \n \u2207<\/m:mo>\n \u2061<\/m:mo>\n u<\/m:mi>\n <\/m:mrow>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n =<\/m:mo>\n f<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n {-\\nabla\\cdot((\\mu+|\\nabla u|^{p-2})\\nabla u)=f}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is considered,\nwhere \u03bc is a given positive number. An anisotropic a posteriori residual-based error estimator is presented. The error estimator is shown to be equivalent, up to higher order terms, to the error in a quasi-norm. The involved constants being independent of \u03bc, the solution, the mesh size and aspect ratio. An adaptive algorithm is proposed and numerical results are presented when \n \n \n \n p<\/m:mi>\n =<\/m:mo>\n 3<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n {p=3}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. From this model problem, we propose a simplified error estimator and use it in the framework of an industrial application, namely a nonlinear Navier\u2013Stokes problem arising from aluminium electrolysis.<\/jats:p>","DOI":"10.1515\/cmam-2022-0205","type":"journal-article","created":{"date-parts":[[2024,6,25]],"date-time":"2024-06-25T17:43:24Z","timestamp":1719337404000},"page":"487-509","source":"Crossref","is-referenced-by-count":1,"title":["Anisotropic Adaptive Finite Elements for a p<\/i>-Laplacian Problem"],"prefix":"10.1515","volume":"25","author":[{"given":"Paride","family":"Passelli","sequence":"first","affiliation":[{"name":"Institute of Mathematics , 27218 EPFL , 1015 Lausanne , Switzerland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0069-5856","authenticated-orcid":false,"given":"Marco","family":"Picasso","sequence":"additional","affiliation":[{"name":"Institute of Mathematics , 27218 EPFL , 1015 Lausanne , Switzerland"}]}],"member":"374","published-online":{"date-parts":[[2024,6,26]]},"reference":[{"key":"2025032910202837691_j_cmam-2022-0205_ref_001","doi-asserted-by":"crossref","unstructured":"M. Ainsworth and J. T. Oden,\nA posteriori error estimation in finite element analysis,\nComput. Methods Appl. Mech. Engrg. 142 (1997), no. 1\u20132, 1\u201388.","DOI":"10.1016\/S0045-7825(96)01107-3"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_002","doi-asserted-by":"crossref","unstructured":"M. Ainsworth, J. Z. Zhu, A. W. Craig and O. C. Zienkiewicz,\nAnalysis of the Zienkiewicz\u2013Zhu a posteriori error estimator in the finite element method,\nInternat. J. Numer. Methods Engrg. 28 (1989), no. 9, 2161\u20132174.","DOI":"10.1002\/nme.1620280912"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_003","doi-asserted-by":"crossref","unstructured":"F. Alauzet and A. Loseille,\nA decade of progress on anisotropic mesh adaptation for computational fluid dynamics,\nComput.-Aided Des. 72 (2016), 13\u201339.","DOI":"10.1016\/j.cad.2015.09.005"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_004","unstructured":"I. Babu\u0161ka, J. Chandra and J. E. Flaherty,\nAdaptive Computational Methods for Partial Differential Equations,\nSociety for Industrial and Applied Mathematics, Philadelphia, 1983."},{"key":"2025032910202837691_j_cmam-2022-0205_ref_005","doi-asserted-by":"crossref","unstructured":"I. Babu\u0161ka, R. Dur\u00e1n and R. Rodr\u00edguez,\nAnalysis of the efficiency of an a posteriori error estimator for linear triangular finite elements,\nSIAM J. Numer. Anal. 29 (1992), no. 4, 947\u2013964.","DOI":"10.1137\/0729058"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_006","doi-asserted-by":"crossref","unstructured":"E. B\u00e4nsch,\nLocal mesh refinement in 2 and 3 dimensions,\nImpact Comput. Sci. Engrg. 3 (1991), no. 3, 181\u2013191.","DOI":"10.1016\/0899-8248(91)90006-G"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_007","doi-asserted-by":"crossref","unstructured":"J. W. Barrett and W. B. Liu,\nQuasi-norm error bounds for the finite element approximation of a non-Newtonian flow,\nNumer. Math. 68 (1994), no. 4, 437\u2013456.","DOI":"10.1007\/s002110050071"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_008","doi-asserted-by":"crossref","unstructured":"L. Belenki, L. Diening and C. Kreuzer,\nOptimality of an adaptive finite element method for the p-Laplacian equation,\nIMA J. Numer. Anal. 32 (2012), no. 2, 484\u2013510.","DOI":"10.1093\/imanum\/drr016"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_009","unstructured":"E. Boey, Y. Bourgault and T. Giordano,\nAnisotropic space-time adaptation for reaction-diffusion problems,\npreprint (2017), https:\/\/arxiv.org\/abs\/1707.04787."},{"key":"2025032910202837691_j_cmam-2022-0205_ref_010","doi-asserted-by":"crossref","unstructured":"Y. Bourgault and M. Picasso,\nAnisotropic error estimates and space adaptivity for a semidiscrete finite element approximation of the transient transport equation,\nSIAM J. Sci. Comput. 35 (2013), no. 2, A1192\u2013A1211.","DOI":"10.1137\/120891320"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_011","doi-asserted-by":"crossref","unstructured":"W. Cao,\nSuperconvergence analysis of the linear finite element method and a gradient recovery postprocessing on anisotropic meshes,\nMath. Comp. 84 (2015), no. 291, 89\u2013117.","DOI":"10.1090\/S0025-5718-2014-02846-9"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_012","doi-asserted-by":"crossref","unstructured":"C. Carstensen,\nAll first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable,\nMath. Comp. 73 (2004), no. 247, 1153\u20131165.","DOI":"10.1090\/S0025-5718-03-01580-1"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_013","doi-asserted-by":"crossref","unstructured":"C. Carstensen and R. Klose,\nA posteriori finite element error control for the p-Laplace problem,\nSIAM J. Sci. 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Phys. 230 (2011), no. 7, 2391\u20132405.","DOI":"10.1016\/j.jcp.2010.11.041"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_017","doi-asserted-by":"crossref","unstructured":"L. Diening and C. Kreuzer,\nLinear convergence of an adaptive finite element method for the p-Laplacian equation,\nSIAM J. Numer. Anal. 46 (2008), no. 2, 614\u2013638.","DOI":"10.1137\/070681508"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_018","doi-asserted-by":"crossref","unstructured":"W. D\u00f6rfler,\nA convergent adaptive algorithm for Poisson\u2019s equation,\nSIAM J. Numer. Anal. 33 (1996), no. 3, 1106\u20131124.","DOI":"10.1137\/0733054"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_019","unstructured":"S. Dubuis,\nAdaptive algorithms for two fluids flows with anisotropic finite elements and order two time discretizations,\nPh.D. Thesis, Ecole polytechnique f\u00e9d\u00e9rale de Lausanne, 2019."},{"key":"2025032910202837691_j_cmam-2022-0205_ref_020","doi-asserted-by":"crossref","unstructured":"S. Dubuis, P. Passelli and M. Picasso,\nAnisotropic adaptive finite elements for an elliptic problem with strongly varying diffusion coefficient,\nComput. Methods Appl. Math. 22 (2022), no. 3, 529\u2013543.","DOI":"10.1515\/cmam-2022-0036"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_021","doi-asserted-by":"crossref","unstructured":"L. Formaggia, S. Micheletti and S. Perotto,\nAnisotropic mesh adaption in computational fluid dynamics: Application to the advection-diffusion-reaction and the Stokes problems,\nAppl. Numer. Math. 51 (2004), no. 4, 511\u2013533.","DOI":"10.1016\/j.apnum.2004.06.007"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_022","doi-asserted-by":"crossref","unstructured":"L. Formaggia and S. Perotto,\nNew anisotropic a priori error estimates,\nNumer. Math. 89 (2001), no. 4, 641\u2013667.","DOI":"10.1007\/s002110100273"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_023","doi-asserted-by":"crossref","unstructured":"L. Formaggia and S. Perotto,\nAnisotropic error estimates for elliptic problems,\nNumer. Math. 94 (2003), no. 1, 67\u201392.","DOI":"10.1007\/s00211-002-0415-z"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_024","doi-asserted-by":"crossref","unstructured":"P. J. Frey and F. Alauzet,\nAnisotropic mesh adaptation for CFD computations,\nComput. Methods Appl. Mech. Engrg. 194 (2005), no. 48\u201349, 5068\u20135082.","DOI":"10.1016\/j.cma.2004.11.025"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_025","doi-asserted-by":"crossref","unstructured":"R. Glowinski and A. Marrocco,\nSur l\u2019approximation, par \u00e9l\u00e9ments finis d\u2019ordre un, et la r\u00e9solution, par p\u00e9nalisation-dualit\u00e9, d\u2019une classe de probl\u00e8mes de Dirichlet non lin\u00e9aires,\nRev. Fran\u00e7. Automat. Inform. Rech. Op\u00e9r. S\u00e9r. Rouge Anal. Num\u00e9r. 9 (1975), no. R-2, 41\u201376.","DOI":"10.1051\/m2an\/197509R200411"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_026","doi-asserted-by":"crossref","unstructured":"O. Gorynina, A. Lozinski and M. Picasso,\nTime and space adaptivity of the wave equation discretized in time by a second-order scheme,\nIMA J. Numer. Anal. 39 (2019), no. 4, 1672\u20131705.","DOI":"10.1093\/imanum\/dry048"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_027","doi-asserted-by":"crossref","unstructured":"G. Kunert,\nAn a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes,\nNumer. Math. 86 (2000), no. 3, 471\u2013490.","DOI":"10.1007\/s002110000170"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_028","doi-asserted-by":"crossref","unstructured":"G. Kunert and S. Nicaise,\nZienkiewicz\u2013Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes,\nM2AN Math. Model. Numer. Anal. 37 (2003), no. 6, 1013\u20131043.","DOI":"10.1051\/m2an:2003065"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_029","doi-asserted-by":"crossref","unstructured":"G. Kunert and R. Verf\u00fcrth,\nEdge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes,\nNumer. Math. 86 (2000), no. 2, 283\u2013303.","DOI":"10.1007\/PL00005407"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_030","unstructured":"P. Laug and H. Borouchaki,\nThe BL2D mesh generator \u2013 Beginner\u2019s guide, user\u2019s and programmer\u2019s manual,\nTechnical Report RT-0194, Institut National de Recherche en Informatique et Automatique (INRIA), Rocquencourt-Le Chesnay, 1996."},{"key":"2025032910202837691_j_cmam-2022-0205_ref_031","doi-asserted-by":"crossref","unstructured":"J.-L. Lions,\nOptimal Control of Systems Governed by Partial Differential Equations,\nGrundlehren Math. Wiss. 170,\nSpringer, New York, 1971.","DOI":"10.1007\/978-3-642-65024-6"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_032","doi-asserted-by":"crossref","unstructured":"W. Liu and N. Yan,\nQuasi-norm local error estimators for p-Laplacian,\nSIAM J. Numer. Anal. 39 (2001), no. 1, 100\u2013127.","DOI":"10.1137\/S0036142999351613"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_033","unstructured":"W. Liu and N. Yan,\nSome a posteriori error estimators for p-Laplacian based on residual estimation or gradient recovery,\nJ. Sci. Comput. 16 (2001), no. 4, 435\u2013477."},{"key":"2025032910202837691_j_cmam-2022-0205_ref_034","doi-asserted-by":"crossref","unstructured":"W. B. Liu and J. W. Barrett,\nQuasi-norm error bounds for the finite element approximation of some degenerate quasilinear elliptic equations and variational inequalities,\nRAIRO Mod\u00e9l. Math. Anal. Num\u00e9r. 28 (1994), no. 6, 725\u2013744.","DOI":"10.1051\/m2an\/1994280607251"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_035","doi-asserted-by":"crossref","unstructured":"A. Loseille and F. Alauzet,\nContinuous mesh framework part I: Well-posed continuous interpolation error,\nSIAM J. Numer. Anal. 49 (2011), no. 1, 38\u201360.","DOI":"10.1137\/090754078"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_036","doi-asserted-by":"crossref","unstructured":"A. Lozinski, M. Picasso and V. Prachittham,\nAn anisotropic error estimator for the Crank\u2013Nicolson method: Application to a parabolic problem,\nSIAM J. Sci. Comput. 31 (2009), no. 4, 2757\u20132783.","DOI":"10.1137\/080715135"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_037","doi-asserted-by":"crossref","unstructured":"S. Micheletti and S. Perotto,\nReliability and efficiency of an anisotropic Zienkiewicz\u2013Zhu error estimator,\nComput. Methods Appl. Mech. Engrg. 195 (2006), no. 9\u201312, 799\u2013835.","DOI":"10.1016\/j.cma.2005.02.009"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_038","doi-asserted-by":"crossref","unstructured":"S. Micheletti, S. Perotto and M. Picasso,\nStabilized finite elements on anisotropic meshes: A priori error estimates for the advection-diffusion and the Stokes problems,\nSIAM J. Numer. Anal. 41 (2003), no. 3, 1131\u20131162.","DOI":"10.1137\/S0036142902403759"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_039","doi-asserted-by":"crossref","unstructured":"J.-M. Mirebeau,\nOptimally adapted meshes for finite elements of arbitrary order and \n \n \n \n W\n \n 1\n ,\n p\n \n \n \n \n W^{1,p}\n \n norms,\nNumer. Math. 120 (2012), no. 2, 271\u2013305.","DOI":"10.1007\/s00211-011-0412-1"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_040","doi-asserted-by":"crossref","unstructured":"M. Picasso,\nAdaptive finite elements for a linear parabolic problem,\nComput. Methods Appl. Mech. Engrg. 167 (1998), no. 3\u20134, 223\u2013237.","DOI":"10.1016\/S0045-7825(98)00121-2"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_041","doi-asserted-by":"crossref","unstructured":"M. Picasso,\nAn anisotropic error indicator based on Zienkiewicz\u2013Zhu error estimator: Application to elliptic and parabolic problems,\nSIAM J. Sci. Comput. 24 (2003), no. 4, 1328\u20131355.","DOI":"10.1137\/S1064827501398578"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_042","doi-asserted-by":"crossref","unstructured":"M. Picasso,\nNumerical study of the effectivity index for an anisotropic error indicator based on Zienkiewicz\u2013Zhu error estimator,\nComm. Numer. Methods Engrg. 19 (2003), no. 1, 13\u201323.","DOI":"10.1002\/cnm.546"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_043","doi-asserted-by":"crossref","unstructured":"M. Picasso,\nAdaptive finite elements with large aspect ratio based on an anisotropic error estimator involving first order derivatives,\nComput. Methods Appl. Mech. Engrg. 196 (2006), no. 1\u20133, 14\u201323.","DOI":"10.1016\/j.cma.2005.11.018"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_044","doi-asserted-by":"crossref","unstructured":"M. Picasso,\nNumerical study of an anisotropic error estimator in the \n \n \n \n \n L\n 2\n \n \u2062\n \n (\n \n H\n 1\n \n )\n \n \n \n \n L^{2}(H^{1})\n \n norm for the finite element discretization of the wave equation,\nSIAM J. Sci. Comput. 32 (2010), no. 4, 2213\u20132234.","DOI":"10.1137\/090778249"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_045","doi-asserted-by":"crossref","unstructured":"S. B. Pope,\nTurbulent flows,\nMeas. Sci. Technol 12 (2001), Paper No. 11.","DOI":"10.1088\/0957-0233\/12\/11\/705"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_046","unstructured":"J. Rochat,\nApproximation num\u00e9rique des coulements turbulents dans des cuves d\u2019\u00e9lectrolyse de l\u2019aluminium,\nPh.D. Thesis, Ecole polytechnique f\u00e9d\u00e9rale de Lausanne, 2016."},{"key":"2025032910202837691_j_cmam-2022-0205_ref_047","doi-asserted-by":"crossref","unstructured":"R. Verf\u00fcrth,\nA posteriori error estimation and adaptive mesh-refinement techniques,\nJ. Comput. Appl. Math. 50 (1994), no. 1\u20133, 67\u201383.","DOI":"10.1016\/0377-0427(94)90290-9"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_048","unstructured":"R. Verf\u00fcrth,\nA Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques,\nWiley-Teubner, New York, 1996."},{"key":"2025032910202837691_j_cmam-2022-0205_ref_049","doi-asserted-by":"crossref","unstructured":"R. Verf\u00fcrth,\nA Posteriori Error Estimation Techniques for Finite Element Methods,\nOxford University, Oxford, 2013.","DOI":"10.1093\/acprof:oso\/9780199679423.001.0001"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_050","doi-asserted-by":"crossref","unstructured":"J. Xu and Z. Zhang,\nAnalysis of recovery type a posteriori error estimators for mildly structured grids,\nMath. Comp. 73 (2004), no. 247, 1139\u20131152.","DOI":"10.1090\/S0025-5718-03-01600-4"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_051","doi-asserted-by":"crossref","unstructured":"O. C. Zienkiewicz and J. Z. Zhu,\nA simple error estimator and adaptive procedure for practical engineering analysis,\nInternat. J. Numer. Methods Engrg. 24 (1987), no. 2, 337\u2013357.","DOI":"10.1002\/nme.1620240206"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_052","doi-asserted-by":"crossref","unstructured":"O. C. Zienkiewicz and J. Z. Zhu,\nThe superconvergent patch recovery and a posteriori error estimates. I. The recovery technique,\nInternat. J. Numer. Methods Engrg. 33 (1992), no. 7, 1331\u20131364.","DOI":"10.1002\/nme.1620330702"},{"key":"2025032910202837691_j_cmam-2022-0205_ref_053","unstructured":"Spatial Corp Headquarters, Broomfield, 3D Precise Mesh, https:\/\/www.3ds.com\/."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0205\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0205\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,29]],"date-time":"2025-03-29T10:25:03Z","timestamp":1743243903000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0205\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,26]]},"references-count":53,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,11,14]]},"published-print":{"date-parts":[[2025,4,1]]}},"alternative-id":["10.1515\/cmam-2022-0205"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2022-0205","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2024,6,26]]}}}