{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T11:14:07Z","timestamp":1760267647721},"reference-count":31,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We consider a mixed finite element method for an obstacle problem with the <jats:italic>p<\/jats:italic>-Laplace differential operator for <jats:inline-formula id=\"j_cmam-2018-0015_ineq_9999_w2aab3b7e1518b1b6b1aab1c14b1b3Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>p<\/m:mi>\n                              <m:mo>\u2208<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">(<\/m:mo>\n                                 <m:mn>1<\/m:mn>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mi mathvariant=\"normal\">\u221e<\/m:mi>\n                                 <m:mo stretchy=\"false\">)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0015_eq_0404.png\" \/>\n                        <jats:tex-math>{p\\in(1,\\infty)}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>,\nwhere the obstacle condition is imposed by using a Lagrange multiplier. In the discrete setting the Lagrange multiplier basis forms a biorthogonal system with the standard finite element basis so that\nthe variational inequality can be realized in the point-wise form.\nWe provide a general a posteriori error estimate for adaptivity and prove an a priori error estimate.\nWe present numerical results for the adaptive scheme (mesh-size adaptivity with and without polynomial degree adaptation) for the singular case <jats:inline-formula id=\"j_cmam-2018-0015_ineq_9998_w2aab3b7e1518b1b6b1aab1c14b1b5Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>p<\/m:mi>\n                              <m:mo>=<\/m:mo>\n                              <m:mn>1.5<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0015_eq_0397.png\" \/>\n                        <jats:tex-math>{p=1.5}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> and the degenerated case <jats:inline-formula id=\"j_cmam-2018-0015_ineq_9997_w2aab3b7e1518b1b6b1aab1c14b1b7Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>p<\/m:mi>\n                              <m:mo>=<\/m:mo>\n                              <m:mn>3<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0015_eq_0399.png\" \/>\n                        <jats:tex-math>{p=3}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>. We also present numerical results on the mesh independency and on the polynomial degree scaling of the discrete inf-sup constant when using biorthogonal basis functions for the dual variable defined on the same mesh with the same polynomial degree distribution.<\/jats:p>","DOI":"10.1515\/cmam-2018-0015","type":"journal-article","created":{"date-parts":[[2019,3,31]],"date-time":"2019-03-31T12:04:02Z","timestamp":1554033842000},"page":"169-188","source":"Crossref","is-referenced-by-count":6,"title":["Higher Order Mixed FEM for the Obstacle Problem of the <i>p<\/i>-Laplace Equation Using Biorthogonal Systems"],"prefix":"10.1515","volume":"19","author":[{"given":"Lothar","family":"Banz","sequence":"first","affiliation":[{"name":"Department of Mathematics , University of Salzburg , Hellbrunner Stra\u00dfe 34, 5020 Salzburg , Austria"}]},{"given":"Bishnu P.","family":"Lamichhane","sequence":"additional","affiliation":[{"name":"School of Mathematical & Physical Sciences , University of Newcastle , University Drive , Callaghan , NSW 2308 , Australia"}]},{"given":"Ernst P.","family":"Stephan","sequence":"additional","affiliation":[{"name":"Institute of Applied Mathematics , Leibniz University Hannover , Welfengarten 1, 30167 Hannover , Germany"}]}],"member":"374","published-online":{"date-parts":[[2018,6,21]]},"reference":[{"key":"2023033110021961968_j_cmam-2018-0015_ref_001_w2aab3b7e1518b1b6b1ab2b2b1Aa","unstructured":"L.  Banz, B. P.  Lamichhane and E. P.  Stephan,\nHigher order FEM for the obstacle problem of the p-Laplacian \u2013 A variational inequality approach,\npreprint, (2017)."},{"key":"2023033110021961968_j_cmam-2018-0015_ref_002_w2aab3b7e1518b1b6b1ab2b2b2Aa","doi-asserted-by":"crossref","unstructured":"L.  Banz and A.  Schr\u00f6der,\nBiorthogonal basis functions in hp-adaptive FEM for elliptic obstacle problems,\nComput. Math. Appl. 70 (2015), no. 8, 1721\u20131742.","DOI":"10.1016\/j.camwa.2015.07.010"},{"key":"2023033110021961968_j_cmam-2018-0015_ref_003_w2aab3b7e1518b1b6b1ab2b2b3Aa","doi-asserted-by":"crossref","unstructured":"L.  Banz and E. P.  Stephan,\nA posteriori error estimates of hp-adaptive IPDG-FEM for elliptic obstacle problems,\nAppl. Numer. Math. 76 (2014), 76\u201392.","DOI":"10.1016\/j.apnum.2013.10.004"},{"key":"2023033110021961968_j_cmam-2018-0015_ref_004_w2aab3b7e1518b1b6b1ab2b2b4Aa","doi-asserted-by":"crossref","unstructured":"L.  Banz and E. P.  Stephan,\nhp-adaptive IPDG\/TDG-FEM for parabolic obstacle problems,\nComput. Math. Appl. 67 (2014), 712\u2013731.","DOI":"10.1016\/j.camwa.2013.03.003"},{"key":"2023033110021961968_j_cmam-2018-0015_ref_005_w2aab3b7e1518b1b6b1ab2b2b5Aa","doi-asserted-by":"crossref","unstructured":"J. W.  Barrett and W.  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":"2023033110021961968_j_cmam-2018-0015_ref_006_w2aab3b7e1518b1b6b1ab2b2b6Aa","doi-asserted-by":"crossref","unstructured":"S.  Bartels and C.  Carstensen,\nAveraging techniques yield reliable a posteriori finite element error control for obstacle problems,\nNumer. Math. 99 (2004), 225\u2013249.","DOI":"10.1007\/s00211-004-0553-6"},{"key":"2023033110021961968_j_cmam-2018-0015_ref_007_w2aab3b7e1518b1b6b1ab2b2b7Aa","doi-asserted-by":"crossref","unstructured":"D.  Braess,\nA posteriori error estimators for obstacle problems\u2013another look,\nNumer. Math. 101 (2005), no. 3, 415\u2013421.","DOI":"10.1007\/s00211-005-0634-1"},{"key":"2023033110021961968_j_cmam-2018-0015_ref_008_w2aab3b7e1518b1b6b1ab2b2b8Aa","doi-asserted-by":"crossref","unstructured":"D.  Braess, C.  Carstensen and R.  Hoppe,\nConvergence analysis of a conforming adaptive finite element method for an obstacle problem,\nNumer. Math. 107 (2007), 455\u2013471.","DOI":"10.1007\/s00211-007-0098-6"},{"key":"2023033110021961968_j_cmam-2018-0015_ref_009_w2aab3b7e1518b1b6b1ab2b2b9Aa","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner and L. R.  Scott,\nThe Mathematical Theory of Finite Element Methods, 3 ed.,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2023033110021961968_j_cmam-2018-0015_ref_010_w2aab3b7e1518b1b6b1ab2b2c10Aa","doi-asserted-by":"crossref","unstructured":"C.  Carstensen and J.  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Gwinner,\nhp-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics,\nJ. Comput. Appl. Math. 254 (2013), 175\u2013184.","DOI":"10.1016\/j.cam.2013.03.013"},{"key":"2023033110021961968_j_cmam-2018-0015_ref_017_w2aab3b7e1518b1b6b1ab2b2c17Aa","doi-asserted-by":"crossref","unstructured":"S.  H\u00fceber and B.  Wohlmuth,\nA primal\u2013dual active set strategy for non-linear multibody contact problems,\nComput. Methods Appl. Mech. Engrg. 194 (2005), no. 27\u201329, 3147\u20133166.","DOI":"10.1016\/j.cma.2004.08.006"},{"key":"2023033110021961968_j_cmam-2018-0015_ref_018_w2aab3b7e1518b1b6b1ab2b2c18Aa","doi-asserted-by":"crossref","unstructured":"G.  Jouvet and E.  Bueler,\nSteady, shallow ice sheets as obstacle problems: Well-posedness and finite element approximation,\nSIAM J. Appl. Math. 72 (2012), no. 4, 1292\u20131314.","DOI":"10.1137\/110856654"},{"key":"2023033110021961968_j_cmam-2018-0015_ref_019_w2aab3b7e1518b1b6b1ab2b2c19Aa","doi-asserted-by":"crossref","unstructured":"R.  Krause, B.  M\u00fcller and G.  Starke,\nAn adaptive least-squares mixed finite element method for the signorini problem,\nNumer. Methods Partial Differential Equations 33 (2017), 276\u2013289.","DOI":"10.1002\/num.22086"},{"key":"2023033110021961968_j_cmam-2018-0015_ref_020_w2aab3b7e1518b1b6b1ab2b2c20Aa","doi-asserted-by":"crossref","unstructured":"A.  Krebs and E.  Stephan,\nA p-version finite element method for nonlinear elliptic variational inequalities in 2D,\nNumer. Math. 105 (2007), no. 3, 457\u2013480.","DOI":"10.1007\/s00211-006-0035-0"},{"key":"2023033110021961968_j_cmam-2018-0015_ref_021_w2aab3b7e1518b1b6b1ab2b2c21Aa","doi-asserted-by":"crossref","unstructured":"B.  Lamichhane and B.  Wohlmuth,\nBiorthogonal bases with local support and approximation properties,\nMath. 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Anal. 43 (2005), no. 1, 127\u2013155.","DOI":"10.1137\/S0036142903432930"},{"key":"2023033110021961968_j_cmam-2018-0015_ref_028_w2aab3b7e1518b1b6b1ab2b2c28Aa","doi-asserted-by":"crossref","unstructured":"N.  Ovcharova and L.  Banz,\nCoupling regularization and adaptive hp-BEM for the solution of a delamination problem,\nNumer. Math. 137 (2017), 303\u2013337.","DOI":"10.1007\/s00211-017-0879-5"},{"key":"2023033110021961968_j_cmam-2018-0015_ref_029_w2aab3b7e1518b1b6b1ab2b2c29Aa","unstructured":"J.  Qin,\nOn the convergence of some low order mixed finite elements for incompressible fluids,\nPh.D. thesis, The Pennsylvania State University, 1994."},{"key":"2023033110021961968_j_cmam-2018-0015_ref_030_w2aab3b7e1518b1b6b1ab2b2c30Aa","doi-asserted-by":"crossref","unstructured":"A.  Veeser,\nEfficient and reliable a posteriori error estimators for elliptic obstacle problems,\nSIAM J. Numer. Anal. 39 (2001), no. 1, 146\u2013167.","DOI":"10.1137\/S0036142900370812"},{"key":"2023033110021961968_j_cmam-2018-0015_ref_031_w2aab3b7e1518b1b6b1ab2b2c31Aa","doi-asserted-by":"crossref","unstructured":"T.  Wick,\nAn error-oriented Newton\/inexact augmented Lagrangian approach for fully monolithic phase-field fracture propagation,\nSIAM J. Sci. 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