{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,7,4]],"date-time":"2023-07-04T14:40:32Z","timestamp":1688481632462},"reference-count":36,"publisher":"Walter de Gruyter GmbH","issue":"3","license":[{"start":{"date-parts":[[2023,2,14]],"date-time":"2023-02-14T00:00:00Z","timestamp":1676332800000},"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":[[2023,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>For the singular integral definition of the fractional Laplacian, we consider an adaptive finite element method steered by two-level error indicators.\nFor this algorithm, we show linear convergence in two and three space dimensions as well as convergence of the algorithm with optimal algebraic rates in 2D, when newest vertex bisection is employed for mesh refinement.\nA key step hereby is an equivalence of the nodal and Scott\u2013Zhang interpolation operators in certain weighted <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>L<\/m:mi>\n                              <m:mn>2<\/m:mn>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0195_ineq_0001.png\" \/>\n                        <jats:tex-math>L^{2}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-norms.<\/jats:p>","DOI":"10.1515\/cmam-2022-0195","type":"journal-article","created":{"date-parts":[[2023,2,13]],"date-time":"2023-02-13T12:36:37Z","timestamp":1676291797000},"page":"603-621","source":"Crossref","is-referenced-by-count":0,"title":["Two-Level Error Estimation for the Integral Fractional Laplacian"],"prefix":"10.1515","volume":"23","author":[{"given":"Markus","family":"Faustmann","sequence":"first","affiliation":[{"name":"Institute of Analysis and Scientific Computing , TU Wien , Wiedner Hauptstr. 8\u201310, 1040 Wien , Austria"}]},{"given":"Ernst P.","family":"Stephan","sequence":"additional","affiliation":[{"name":"Institute of Applied Mathematics , Leibniz University Hannover , Welfengarten 1, 30167 Hannover , Germany"}]},{"given":"David","family":"W\u00f6rg\u00f6tter","sequence":"additional","affiliation":[{"name":"Institute of Analysis and Scientific Computing , TU Wien , Wiedner Hauptstr. 8\u201310, 1040 Wien , Austria"}]}],"member":"374","published-online":{"date-parts":[[2023,2,14]]},"reference":[{"key":"2023070413382873658_j_cmam-2022-0195_ref_001","doi-asserted-by":"crossref","unstructured":"G. Acosta, F. M. Bersetche and J. P. Borthagaray,\nA short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian,\nComput. Math. Appl. 74 (2017), no. 4, 784\u2013816.","DOI":"10.1016\/j.camwa.2017.05.026"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_002","doi-asserted-by":"crossref","unstructured":"G. Acosta and J. P. Borthagaray,\nA fractional Laplace equation: Regularity of solutions and finite element approximations,\nSIAM J. Numer. Anal. 55 (2017), no. 2, 472\u2013495.","DOI":"10.1137\/15M1033952"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_003","doi-asserted-by":"crossref","unstructured":"M. Ainsworth and C. Glusa,\nAspects of an adaptive finite element method for the fractional Laplacian: A priori and a posteriori error estimates, efficient implementation and multigrid solver,\nComput. Methods Appl. Mech. Engrg. 327 (2017), 4\u201335.","DOI":"10.1016\/j.cma.2017.08.019"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_004","doi-asserted-by":"crossref","unstructured":"M. Aurada, M. Feischl, T. F\u00fchrer, M. Karkulik and D. Praetorius,\nEnergy norm based error estimators for adaptive BEM for hypersingular integral equations,\nAppl. Numer. Math. 95 (2015), 15\u201335.","DOI":"10.1016\/j.apnum.2013.12.004"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_005","doi-asserted-by":"crossref","unstructured":"R. E. Bank and A. Weiser,\nSome a posteriori error estimators for elliptic partial differential equations,\nMath. Comp. 44 (1985), no. 170, 283\u2013301.","DOI":"10.1090\/S0025-5718-1985-0777265-X"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_006","doi-asserted-by":"crossref","unstructured":"A. Bespalov, D. Praetorius and M. Ruggeri,\nTwo-level a posteriori error estimation for adaptive multilevel stochastic Galerkin finite element method,\nSIAM\/ASA J. Uncertain. Quantif. 9 (2021), no. 3, 1184\u20131216.","DOI":"10.1137\/20M1342586"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_007","doi-asserted-by":"crossref","unstructured":"P. Binev, W. Dahmen and R. DeVore,\nAdaptive finite element methods with convergence rates,\nNumer. Math. 97 (2004), no. 2, 219\u2013268.","DOI":"10.1007\/s00211-003-0492-7"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_008","doi-asserted-by":"crossref","unstructured":"J. P. Borthagaray and R. H. Nochetto,\nBesov regularity for the Dirichlet integral fractional Laplacian in Lipschitz domains,\nJ. Funct. Anal. 284 (2023), no. 6, Paper No. 109829.","DOI":"10.1016\/j.jfa.2022.109829"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_009","doi-asserted-by":"crossref","unstructured":"C. Bucur and E. Valdinoci,\nNonlocal Diffusion and Applications,\nLect. Notes Unione Mat. Ital. 20,\nSpringer, Cham, 2016.","DOI":"10.1007\/978-3-319-28739-3"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_010","doi-asserted-by":"crossref","unstructured":"L. Caffarelli and L. Silvestre,\nAn extension problem related to the fractional Laplacian,\nComm. Partial Differential Equations 32 (2007), no. 7\u20139, 1245\u20131260.","DOI":"10.1080\/03605300600987306"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_011","doi-asserted-by":"crossref","unstructured":"C. Carstensen, M. Feischl, M. Page and D. Praetorius,\nAxioms of adaptivity,\nComput. Math. Appl. 67 (2014), no. 6, 1195\u20131253.","DOI":"10.1016\/j.camwa.2013.12.003"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_012","doi-asserted-by":"crossref","unstructured":"J. M. Cascon, C. Kreuzer, R. H. Nochetto and K. G. Siebert,\nQuasi-optimal convergence rate for an adaptive finite element method,\nSIAM J. Numer. Anal. 46 (2008), no. 5, 2524\u20132550.","DOI":"10.1137\/07069047X"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_013","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet,\nThe Finite Element Method for Elliptic Problems,\nStud. Math. Appl. 4,\nNorth-Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_014","doi-asserted-by":"crossref","unstructured":"E. Di Nezza, G. Palatucci and E. Valdinoci,\nHitchhiker\u2019s guide to the fractional Sobolev spaces,\nBull. Sci. Math. 136 (2012), no. 5, 521\u2013573.","DOI":"10.1016\/j.bulsci.2011.12.004"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_015","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":"2023070413382873658_j_cmam-2022-0195_ref_016","doi-asserted-by":"crossref","unstructured":"C. Erath, S. Ferraz-Leite, S. Funken and D. Praetorius,\nEnergy norm based a posteriori error estimation for boundary element methods in two dimensions,\nAppl. Numer. Math. 59 (2009), no. 11, 2713\u20132734.","DOI":"10.1016\/j.apnum.2008.12.024"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_017","doi-asserted-by":"crossref","unstructured":"C. Erath, G. Gantner and D. Praetorius,\nOptimal convergence behavior of adaptive FEM driven by simple \n                  \n                     \n                        \n                           (\n                           \n                              h\n                              \u2212\n                              \n                                 h\n                                 \/\n                                 2\n                              \n                           \n                           )\n                        \n                     \n                     \n                     (h-h\/2)\n                  \n               -type error estimators,\nComput. Math. Appl. 79 (2020), no. 3, 623\u2013642.","DOI":"10.1016\/j.camwa.2019.07.014"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_018","unstructured":"M. Faustmann, C. Marcati, J. M. Melenk and C. Schwab,\nExponential convergence of \n                  \n                     \n                        \n                           h\n                           \u2062\n                           p\n                        \n                     \n                     \n                     hp\n                  \n               -FEM for the integral fractional Laplacian in polygons, preprint (2022), https:\/\/arxiv.org\/abs\/2209.11468."},{"key":"2023070413382873658_j_cmam-2022-0195_ref_019","doi-asserted-by":"crossref","unstructured":"M. Faustmann, C. Marcati, J. M. Melenk and C. Schwab,\nWeighted analytic regularity for the integral fractional Laplacian in polygons,\nSIAM J. Math. Anal. 54 (2022), no. 6, 6323\u20136357.","DOI":"10.1137\/21M146569X"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_020","doi-asserted-by":"crossref","unstructured":"M. Faustmann, J. M. Melenk and D. Praetorius,\nQuasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian,\nMath. Comp. 90 (2021), no. 330, 1557\u20131587.","DOI":"10.1090\/mcom\/3603"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_021","doi-asserted-by":"crossref","unstructured":"S. A. Funken and A. Schmidt,\nAdaptive mesh refinement in 2D\u2014an efficient implementation in Matlab,\nComput. Methods Appl. Math. 20 (2020), no. 3, 459\u2013479.","DOI":"10.1515\/cmam-2018-0220"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_022","doi-asserted-by":"crossref","unstructured":"T. Gantumur,\nConvergence rates of adaptive methods, Besov spaces, and multilevel approximation,\nFound. Comput. Math. 17 (2017), no. 4, 917\u2013956.","DOI":"10.1007\/s10208-016-9308-x"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_023","unstructured":"H. Gimperlein, E. P. Stephan and J. \u0160to\u010dek,\nCorner singularities for the fractional Laplacian and finite element approximation, preprint (2021), http:\/\/www.macs.hw.ac.uk\/~hg94\/corners.pdf."},{"key":"2023070413382873658_j_cmam-2022-0195_ref_024","doi-asserted-by":"crossref","unstructured":"G. Grubb,\nFractional Laplacians on domains, a development of H\u00f6rmander\u2019s theory of \ud835\udf07-transmission pseudodifferential operators,\nAdv. Math. 268 (2015), 478\u2013528.","DOI":"10.1016\/j.aim.2014.09.018"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_025","doi-asserted-by":"crossref","unstructured":"J. Gwinner and E. P. Stephan,\nAdvanced Boundary Element Methods: Treatment of Boundary Value, Transmission and Contact Problems,\nSpringer Ser. Comput. Math. 52,\nSpringer, Cham, 2018.","DOI":"10.1007\/978-3-319-92001-6"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_026","doi-asserted-by":"crossref","unstructured":"M. Karkulik, D. Pavlicek and D. Praetorius,\nOn 2D newest vertex bisection: Optimality of mesh-closure and \n                  \n                     \n                        \n                           H\n                           1\n                        \n                     \n                     \n                     H^{1}\n                  \n               -stability of \n                  \n                     \n                        \n                           L\n                           2\n                        \n                     \n                     \n                     L_{2}\n                  \n               -projection,\nConstr. Approx. 38 (2013), no. 2, 213\u2013234.","DOI":"10.1007\/s00365-013-9192-4"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_027","doi-asserted-by":"crossref","unstructured":"M. Kwa\u015bnicki,\nTen equivalent definitions of the fractional Laplace operator,\nFract. Calc. Appl. Anal. 20 (2017), no. 1, 7\u201351.","DOI":"10.1515\/fca-2017-0002"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_028","doi-asserted-by":"crossref","unstructured":"M. Maischak, P. Mund and E. P. Stephan,\nAdaptive multilevel BEM for acoustic scattering,\nComput. Methods Appl. Mech. Engrg. 150 (1997), 351\u2013367.","DOI":"10.1016\/S0045-7825(97)00081-9"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_029","doi-asserted-by":"crossref","unstructured":"P. Mund and E. P. Stephan,\nAn adaptive two-level method for the coupling of nonlinear FEM-BEM equations,\nSIAM J. Numer. Anal. 36 (1999), no. 4, 1001\u20131021.","DOI":"10.1137\/S0036142997316499"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_030","doi-asserted-by":"crossref","unstructured":"P. Mund, E. P. Stephan and J. Wei\u00dfe,\nTwo-level methods for the single layer potential in \n                  \n                     \n                        \n                           R\n                           3\n                        \n                     \n                     \n                     {\\mathbf{R}}^{3}\n                  \n               ,\nComputing 60 (1998), no. 3, 243\u2013266.","DOI":"10.1007\/BF02684335"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_031","doi-asserted-by":"crossref","unstructured":"C.-M. Pfeiler and D. Praetorius,\nD\u00f6rfler marking with minimal cardinality is a linear complexity problem,\nMath. Comp. 89 (2020), no. 326, 2735\u20132752.","DOI":"10.1090\/mcom\/3553"},{"key":"2023070413382873658_j_cmam-2022-0195_ref_032","doi-asserted-by":"crossref","unstructured":"D. Praetorius, M. Ruggeri and E. P. Stephan,\nThe saturation assumption yields optimal convergence of two-level adaptive BEM,\nAppl. Numer. 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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"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0195\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0195\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,4]],"date-time":"2023-07-04T14:18:20Z","timestamp":1688480300000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0195\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,14]]},"references-count":36,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,12,7]]},"published-print":{"date-parts":[[2023,7,1]]}},"alternative-id":["10.1515\/cmam-2022-0195"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2022-0195","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,14]]}}}